Carbon Nanotubes: Basic Concepts and Physical Properties

S. Reich, C. Thornsen, J. Maultzsch

Carbon Nanotubes Basic Concepts and Physical Properties

S. Reich, C. Thornsen,J. Maultzsch

Carbon Nanotubes Basic Concepts and Physical Properties

W I LEYVCH

WILEY-VCH Verlag GmbH & CO.KGaA

Authors Prof Christian Thornsen Technische Universillt Berlin, Germany

Dr. Stephanie Xeich [Jniversrty of Cambridge, UK Dipl. Phys. J a n k i Muulczsch Technische Univcrsitat Bcrlin, Germany

Cover Picture The cover shows an electronic wave function of a (19,O) nanotube; white and blue are for different sign.The background is a contour plot ot the conduction band in the graphenc Brillouin zone. First Reprint 2004

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Preface

This book has evolved from a number of years of intensive research on carbon nanotubes. We feel that the knowledge in the literature in the last five years has made a significant leap forward to warrant a comprehensive presentation, Large parts of the book are based on the Ph.D. thesis work of Stephanie Reich and Janina Maultzsch. All of us benefited much from a close scientific collaboration with the group of Pablo Ordej6n at the Institut de Cibncia de Materials dc Barcelona, Spain, on density-functional theory. Many of the results presented in this book would not have been obtained without him; the band-structure calculations we show were performed with the program Sicsta, which he co-authored. We learnt much of the theory of line groups from an intense scientific exchange with the group of Milan Damnjanovik, Faculty of Physic3, Belgrade, Serbia and Monte Negro. Christian Thomsen thanks Manuel Cardona from the Max-Planck Institut fiir Festkorperforschung in Stuttgart, Germany, for introducing him to the fascinating topic of Raman scattering in solids and for teaching him how solid-state physics concepts can be derived from this technique. We acknowlegde the open and intense discussions with many colleagues at physics meetings and workshops, in particular the Krchberg meetings organized by Hans Kuzmany, Wicn, Austria, for many years and thc Nanotcch series of confcrcnces. Stephanie Reich thanks the following bodies for their financial support while working on this book, the Berlin-Brandenburgische Akademie der Wissenschaften, Berlin, Germany, the Oppenheimer Fund, Cambridge, UK, and Newnham College, Cambridge, UK. Janina Maultzsch acknowledges funding from the Deutsche Forschungsgemeinschaft. Marla Machhn, Peter Rafailov, Sabine Bahrs, Ute Habocck, Harald Schecl, Michacl Stoll, Matthias Dworzak, Riidcgcr Kiihler (Pisa, Italy) gave us serious input by critically reading various chapters of the book. Their suggestions have made the hook clearer and better. We thank them and all other members of the research group at the Technische Universitat Berlin, who gave support to the research on graphite and carbon nanotubes over the years, in particular, Heiner Perls, Bernd Scholer, Sabine Morgner, and Marianne Heinold. Michael Stoll compiled the index. We thank Vera Palmer and Ron Schulz from Wiley-VCH for their support.

Stephanie Reich Berlin, October 2003

Christian Thomsen

Janina Maultzsch

Contents

Preface

v

1 Introduction

1

2 Structure and Symmetry 2.1 Structure of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Symmetry of Single-walled Carbon Nanotubes . . . . . . . . . . . . . . . . 2.3.1 Symmetry Operations . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Symmetry-based Quantum Numbers . . . . . . . . . . . . . . . . . . 2.3.3 Irreducible representations . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Phonon Symmetries in Carbon Nanotubes . . . . . . . . . . . . . . . 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Electronic Properties of Carbon Nanotubes

3 3 9 12 12 15 I8 21 27 30

Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3I . I Tight-binding Description of Graphcne . . . . . . . . . . . . . . . . Zone-folding Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Experimental Verifications of the DOS . . . . . . . . . . . . . . . . . Beyond Zone Folding - Curvature Effects . . . . . . . . . . . . . . . . . . . 3.4.1 Secondary Gaps in Metallic Nitnotubes . . . . . . . . . . . . . . . . 3.4.2 Rehybridization of the cr and 7c States . . . . . . . . . . . . . . . . . Nanotube Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Low-energy Properties . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Visible Energy Range . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 33 41 44 47 50 50 53 60 60 62 64

4 Optical Properties 4.1 Absorption and Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Selection Rules and Depolarization . . . . . . . . . . . . . . . . . . 4.2 Spectra of Isolated Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Photoluminescence Excitation - (nl,n z ) Assignment . . . . . . . . . . . . . 4.4 4-A-diameter Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 68 72 73 77

3.1 3.2 3.3

3.4

3.5

3.6

...

v~ll

Contents

4.5 Bundles of Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Excited-state Carrier Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Electronic Transport

79 80 83

5.1 Room-temperature Conductance of Nanotubes . . . . . . . . . . . . . . . . . 5.2 Electron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 CoulombBlockade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 LuttingerLiquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 88 93 96 99

6 Elastic Properties 6.1 Continuum Model of Isolated Nanotubes . . . . . . . . . . . . . . . . . . . . 6.I .1 Abinitio. Tight.binding. and Force-constants Calculations . . . . . . 6.2 Pressure Dependence of the Phonon Frequencies . . . . . . . . . . . . . . . 6.3 Micro-mechanical Manipulations . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 105 107 111 114

7 Rarnan Scattering 7.1 Raman Basics and Selection Rules . . . . . . . . . . . . . . . . . . . . . . . 7.2 Tensor Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Polarized Measurements . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Raman Measurements at Large Phonon q . . . . . . . . . . . . . . . . . . . 7.4 Double Resonant Raman Scattering . . . . . . . . . . . . . . . . . . . . . . 7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - .

115 115 119 121 123 126 133

8 Vibrational Properties 135 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8.2 Radial Breathing Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.2.1 The REM in Isolated and Bundled Nanotubes . . . . . . . . . . . . . 142 8.2.2 Double-walled Nanotubes . . . . . . . . . . . . . . . . . . . . . . . 149 8.3 The Defect-induced D Mode . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.3.1 The D Mode in Graphite . . . . . . . . . . . . . . . . . . . . . . . . 153 8.3.2 The D Mode in Carbon Nanotubes . . . . . . . . . . . . . . . . . . . 154 8.4 Symmetry of the Raman Modes . . . . . . . . . . . . . . . . . . . . . . . . 158 8.5 High-energy Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.5.1 Raman and Infrared Spectroscopy . . . . . . . . . . . . . . . . . . . 162 8.5.2 Metallic Nanotuhes . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.5.3 Single- and Double-resonanceInterpretation . . . . . . . . . . . . . 172 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8.7 What we Can Learn from the Raman Spectra of Singlc-walled Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Appendix A Character and Correlation Tables of Graphene Appendix B Raman Intensities in Unoriented Systems

181

Contents

Appendix C Fundamental Constants

Bibliography Index

1 Introduction

The physics of carbon nanotubes has rapidly evolved into a research field since their discovery by lijima in multiwall for111in 1991 and as single-walled tubes two years later. Since then, theoretical w d experimental studies in different fields, such as mechanics, optics, and electronics have focused on both the fundamental physical properties and on the potential applications of nanotubes. In all fields there has been substantial progrcss over the last decade, the first actual applications appearing on the market now. We prescnt a consistent picture of experimental and thcoreiical studies of carbon nanotubes and offer the reader insight into aspects that are not only applicable to carbon nanotubes but are uscrul physical concepts, in particular, in one-dimensional systems. The book is intended for graduate students and researchers interested in a comprehensive introduction and review of theoretical and experimental concepts in carbon-nanotube research. Emphasis is put on introducing the physical conccpts that frequently differ from common understanding in solidstate physics because of the one-dimensional nature of carbon nanotubes. The two focii of the book, electronic and vihrational properties of carbon nanotubes, rely on a basic understanding of the symmetry of nanotubcs, and we show how symmetry-related techniques can be applied to one-dimensional systems in general. Preparation of nanotubes is not treated in this book, Tor an overview we refer the reader to excellent articles, e.g., Seo rt nl.ll.'l on CVD-related processes. Wc also do not trcal multiwall carbon nanotubes, because dimensionality affects their physical properties to be much closer to those of graphite. Nevcttheless, for applications of carbon nanotuhes they are extrcmely ~] valuable, and we refer to the literature for reviews on this topic, e.g., Ajayan and ~ h o u [ ' . for more information on the topic. The textbook Fundamentals of Semiconductors by Yu and Cardona['." and the series of volumes on Light-Scattcring in ~ o l i d d41 ' was most helpful in developing several chapters in this book. We highly recommend thcse books for rurther reading and for gaining a more basic understanding or some of the advanced conccpts presented hcrc when needcd. There are also a number of excellent books on various topics related to carbon-nanotube research and applications that have appeared bcfore. We mention the volume by Dresselhaus et d.,[' the book by Saito eta(.,' 1.61 thc book by ~ a r r i s [71' and thc collection of articles that was edited by Drcsselhaus ~t They offer valuable introductions and overvicws to a number of carbon nanotube topics not treated hcre. Beginning with the structure and symmetry properties of carbon nanotubcs (Chap. 2), to which many results are intimately connectcd, we present thc electronic band structure of single isolated tubes and of nanotube bundles as one of the two focii of this book in Chap. 3. The optical and transport properties of carbon nanotuhes arc then treated on the basis oS the

2

1 Introduction

electronic hand structure in the optical range and near the Fermi level (Chaps. 4 and 5). We introduce the reader to thc elastic properties of nanotubes in Chap. 6 and lo basic concepts in Raman scattering, as needed in the book, in Chap. 7. The carbon-atom vibrittions are related to the electronic band structure through single and double resonances and constitute the second main focus. We treat the dynamical properties of carbon nanotuhes in Chap. 8, summarizing what we rccl can be learnt from Raman spectroscopy on nanotubes.

2 Structure and Symmetry

Carbon nanotubes are hollow cylinders of graphite sheets. Thcy can be looked at as single molecules, regarding their small size ( w nrn in diameter and -- prn length), or as quasi-one dimensional crystals with translational periodicity along the tube axis. There are infinitely many ways to roll a sheet into a cylinder, resulting in different diameters and microscopic structures of the tubes. These are defined by thc chiral angle, the angle of the hexagon helix around the tube axis. Some properties of carbon nanotubes can be explained within a rnacroscopic model of an homogeneous cylinder (see Chap. 6); whereas others depend crucially on the microscopic structure of thc tubes. The latter include, for instance, thc electronic band structure, in particular, their metallic or semiconducting nature (see Chap. 3). The fairly complex microscopic structure with tens to hundreds o C atoms in the unit cell can be described in a vcry general way with the hclp of the nanotube symmetry. This greatly simplifies calculating and understanding physical properties like oplical absorption, phonon eigenvectors, and electron-phonon coupling. In this chapter we first dcscribe the geometric structure of carbon nanotubes and the construction of their Brillouin zone in relation to lhal of graphite (Sect. 2.1). In Sect. 2.2 we give an overview of experimental methods to determine the atomic structure of carbon nanotubes. The symmetry propertieq of singlc-walled tubes are presented in Sect. 2.3. We cxplain how to obtain the entire tube of a given chirality from one single carbon atom by applying the symmetry operations. Furthermore, we givc an introduction to the theory of line groups or carbon nanotubc~,[~.'I and explain the quantum numbers, irreducible representations, and their notation. Finally, we show how to use a graphical method of group projectors to dcrive normal modes from ryrnrnetry (Sect. 2.3.4), and present the phonon symmetries in nanotubes.

2.1 Structure of Carbon Nanotubes A tube made of a single graphitc layer rolled up into a hollow cylinder is called a single-walled nanotube (SWNT); a tube comprising scveral, concentrically arranged cylinders is rcfcrred to as a multiwall tube (MWNT). Single-walled nanotubes, as typically investigated in the work presented hcre, are produced by laser ablation, high-pressure CO conversion (HiPCO), or the arc-discharge technique and have a Gaussian distribution of diameters d with mean diameters do E 1.0 - 1.5nm.L2 21-12.51 The chiral angles [Eq, (2.I)] are evenly distributed.[2." Singlewalled tubes form hexagonal-packed bundles during the growth process. Figure 2.1 shows a transmission electron microscopy image of such a bundle. The wall-to-wall distance hetwccn two tubes is in the same range as the interlaycr distance in graphite (3.41 A). Multiwall nanotubes have similar lengths to single-walled tubes, but much larger diameters. Their inner m d

Figure 2.1: High-resolution transmission electron micrc)scopy (TEM) picture of a bundle of single-walled nanotubes. The

hexagonal packing is nicely seen in the edge-on picture. Taken from Ref. 12.21.

outer diameters are around 5 and 100nm, respectively, corrcsponding to F= 30 coaxial tubes. Confinement effccts are expected to be less dominant than in singlc-walled tubes, because of the large circumference. Many of the properties of multiwall tubes are already quite close to graphite. While the multiwall nanotubes have a wide range of application, they are less well defined from their structural and hence electronic properties due to the many possible number of layers. Because the microscopic structure of carbon nanotubes is closely related to gaphene1,the tubes are usually labeled in terms of the graphcne lattice vectors. In the following sections we show that by this rcference to graphene many properties of carbon nanotubes can be derived. Figure 2.2 shows the graphene honeycomb lattice. The unit cell is spanned by the two vectors a1 and a2 and contains two carbon atoms at the positions ( a l -tan)and $ ( a l a 2 ) , whcrc the basis vectors of length lal I = la21 = a0 = 2.461 form an angle of 60'. In carbon nanotubcs, the graphene sheet is rolled up in such a way that a graphene lattice vector c = nlal n2az becomes the circumfcrcnce of the tube. This circumferential vector c, which is usually denoted by the pair of integers (n1 , n z ) , is called the chiral vector and uniquely defines ii particular tube. We will scc below that many properties of nanotubes, like thcir electronic band struclurc or the spatial symmetry group, vary dramatically with the chiral vector, cvcn for tubes with similar diameter and direction of the chiral vector. For example, the (10,10) tube contains 40 atoms in the unit cell and is metallic; the close-by (10,9) tube with 1084 atoms in the unit cell is a semiconducting tube. In Fig. 2.2, the chiral vector c = 8al +4a2 of an (8,4) tube is shown. The circles indicate the four points on the chiral vector that are lattice vectors of graphenc; the first and the last circle coincide if the sheet is rolled up. The numbcr of lattice points on the chiral vector is given by the greatest common divisor n of (nl,n 2 ) , sincc c = n ( n l /n . a1 n 2 / n . a2) = n . c' is a multiple of another graphene latlicc vector c'. The direction of the chiral vector is measured by the chiral angle 8, which is defined as the angle between a1 and c. The chiral angle 8 can be calculated frotn

A

4

+

+

+

For each tube with 8 between 0" and 30" an equivalent tube with 8 between 30" and 60" is found, but the helix of graphene lattice points around the tube changes from right-handed lo Icft-handed. Because of the six-fold rotational symmetry of graphene, to any other chiral vector an equivalent one exihth with 0 60". We will hence restrict ourselves to the case


2n/3 have even parity under oh, states with m < 2n/3 transform odd, and vice versa for the valence-band states, where n is the index of a lube (n,O).z-polarized transitions are always allowed between bands with the same m as indexed by the subscript in the irreducible representation, 13 -+ 13, 14 + 14, 15 415, etc. Transitions polarized as XJ cannot change the ~h parity and involve in zig-zag tubes the first two valence and conduction bands only, otherwise the mirror parity is violated. Transitions 13 + 12 or 12 -+ 13 are allowed, while 13 + 14 is forbidden2 (the parity changes).1451314." For zig-zag nanotuhes these selection rules are strict, for the armchair and chiral ones not, but it turns out that the corresponding matrix elements are small. The transition energies in nanotubes have a systematic diameter dependence when looked I I d , the isotropic approximation, at in certain approximations. In the simplest picture E and plotted versus l / d the energies are straight lines. In such a plot, chirality obviously plays no role. Frequently in nanotubes research the first approximation including chirality effects, the nearest-neighbor tight-binding approximation (Sect 3.1.1) is used; such a plot 2 ~ o t that e in general the xy-allowed transitions need not be the two closest to the Fermi level as 111 Rg. 4.1; In rmall tuhes there may he dcv~atlons.The rule is that XJ trmsitions are allowed if rn < 2n/3 and m - 1 5 2 n / 3 .

70

4

Opticul Properties

Figure 4.2: Transition energies of chiral nanotubes as a function of diameter of the nanotube calculated in the third-order tight-binding approximation with thc energies fitted to ah-initio results (Table 3.2). The visible range is highlighted in gray. Compared to the first-order approximation that Kataura et uLL471 used to calculate the energies, they are more spread out. The first metallic band in the visible can barely be distinguished from the higherlying band. Solid symbols are semiconducting tubes, open ones are metallic tubes.

Tube diameter (nm) is commonly referred to as a Kataura plot. Kataura et al.[4.71 first showed how the l/d average behavior splits into different values for different chiralities, see, e.g., Fig. 8.8. We show in Fig. 4.2 a plot of the transition energies versus diameter in the third-nearest neighbor approximation (Sect. 3.1.1 j. In this plot, which also shows the 1/ d behavior on average, the transition energies for specific chiralities are more accurate, according to the improved expression used to calculate them. In the visible energy region, where many experiments are performed, the energies are more spread out than in the first-order approximation; a distinction between the first metallic and the higher-lying semiconducting tubes is barely possible. Higher semiconducting bands start to blur already in the visible, at higher energies they are quite mixed. Related to the selection rules is the suppression of the perpendicular optical absorption by depolarization (often called antenna effectj.[4.81-14.101In the static approximation we can understand this effect by considering a long and narrow cylinder in an applied electric field. In an infinitely long cylinder polarization charges are not induced for a field along the z-axis, and the electric field Ellequals the external field E . For an external field applied in x- or y-direction, however, charges are induced on the cylinder walls. The resulting polarization vector P opposes the external field and reduces the electric field E l , see Fig. 4.3. This effect was first described for carbon nanotubes by Ajiki and ~ n d 0 . L ~ "Later Benedict et derived the dependence of the depolarization on the radius of the cylinder. They find that the Figure 4,3: (a) For an cxternal electric field parallel to the zaxis there is no induced polarization and the total parallel electric field is equal to the external field. (b) Perpendicular to the tube axis the electric field is reduced due to [he polarization induced on the cylinder's surface. In carbon nanotubes the locally induced polarization almost completely cancels the perpendicular electric field.

I

I

4.1 Absorption and Emission

Figure 4.4: Real part of the parallel and perpendicular conductivity in units of e 2 / h for a (50,-10) [equivalent to a (40,10)] tube. In the upper curve the parallel conductivity (.- absorption) is shown; it is unaffected by depolarization. In the lower panel the peak at low energies is almost entircly screened out due to the depolarimtion (unscreened response - dotted, screened - full line). Note also the different absolute energies of thc peaks due to the different absorption selection rules for light parallel and perpendicular to the tubc axis. The full energy axis corrcsponds to 2.7 eV. After Ref. [4.101.

71

,, $

8

tube (50,-10) $-

3.

2 o.

"""

0 25

0 SO

ti

6

b

2c

screened polarizability per unit length of the nanotube in the perpendicular direction is

where d is the diameter of the cylinder, and %,J is the unscreened polarizability. For small diameters the polarizability in the perpendicular direction thus becomes screened and the optical transitions - even if allowed in principle - will not be seen in experiments. Combined with a tight-binding model the electrostatic argument yields a response to perpendicular excitation on average about 5 times weaker than for parallel excitation. ' ~ ] extensively the interaction of nanotubes with optical radiation, Tasaki et U Z . [ ~ ~studied including the depolarixation due to the shape of a nanotube. They calculated, also within a tight-binding model, the optical absorption, the optical activity, and circular dichroism due to the chirality of the nanotubes. Thcy corrected the perpendicular field with a frequencydependent factor

due to the depolarization, where'5 is the surface conductivity along the peripheral direction. Note that the complex conductivity is related to the dielectric constant by Z- = 1 4 z i & / w . For a (40,lO)tube, the calculation shows nicely how the parallel absorption is not affected by screening, see Fig. 4.4. However, in the perpendicular case the difference, in particular for small transition energies, is remarkable: at low energies almost all absorption is screened out, the lowest unscreened absorption can no longer be identified in the plot. A consequence of the depolarization of the external electric field is that absorption-related experiments on randomly oriented nanotubes are completely governed by their properties for z-polarization. This has been shown for narrow nanotubes aligned in zeolite crystals, which consist of regularly arranged channels with a diameter of only 10.1 A. Li et al.I4 121 were able

+

72

4

Optical Properties

Figure 4.5: Transmission through a m a l l nanotuhctillcd zeolite crystal with the electric-field vector pclpendlcular (upper) and parallel (lower) t o thc' z-axis. The depolari7ation effect is secn in the transparent region of the upper imagc. The cryhtals arc about 200pm long. From Ref. [4.11].

to grow narrow carbon nanotubes inside such channels. They found from transmission electron microscopy predominantly tubes with a diameter d = 4 hence three types of nanotubes, the (5,0),(4,2), and (3,3) are taken to be present in a zeolite channel. In Fig. 4.5 we show a picture of nanotuhes grown inside zeolite crystals taken with light polarized perpendicular and parallel to the z-axis. In the upper part of the figure there is only little absorption, while the lower one appears black, hence all of the light is absorbed.[4 lo]

A;

4.2

Spectra of Isolated Tubes

When studying the optical properties one is often hampered by the bundling of the tubes. The luminescence efficiency, e.g., is greatly reduced due to rapid transfer processes from semiconducting to metallic tubes, wherc nun-radiative recombination occurs, see also Sect. 4.6. This is why reports of luminescence have appeared only long after the discovery of nanotubes. reported the preparation of isolated nanotubes in which they were O'Connell et able to prevent rebundling. They prepared such isolated tubes by first ultrasonicating raw nanotube products obtained from the HiPCo process14 141 and dispersing them in an aqueous sodium dodecyl sulfate (SDS) surfactant. The micelles prevented the tubes from reforming bundles. The left panel in Fig. 4.6 shows an isolated nanotube and a small bundle covered by surfactants. To separate the isolated tubes from the remaining bundles O'Connell et 01. L4.l31 made use of the differcnt density and weight of single and bundled tubes. The samples were centrifuged and the upper 80% of the supernatant decanted. The resulting micelle-suspended nanotube solutions had a typical mass concentration of 20mgfl. The isolated nanotubes had richly structured absorption spectra and showed strong photoluminescence. Figure 4.6 shows the optical spectra in the near-infrared for tubes with a diameter of 0.7 to 1.1 nm, typical for the HiPCo process used for the starting material. The isolated tubes show narrow absorption lines and photoluminescence with no shift between absorption and emission. From predictions of wherc the singularities in such nanotubes lie, transitions to and from the first excited state of semiconducting tubes can be identified in the spectra, The photoluminescence was excited with haI = 2.3 eV, which is much higher than the first excited state; the absorption is thus into the second or higher band of the nanotube, and the carriers relax quickly to the lowest excited state before recombining. There are several reasons why the emitted light does not originate from the coating of the tubes. For one, the detected emission energy matches quite well the expected nanotube band gap. Secondly, the emission was also observed in unprocessed nanotubes from free-standing isolated tubes grown on small pillars by chemical vapor deposition ( c v D ) . [ ~ . ~ ~ ~

4.3 Photolumin~srenceExcitation - ( n l ,n2)Assignment

Frequency (cm-I)

Figure 4.6: (left) Model of an isolated nanotube and a 7-tube bundle covered with a layer of SDS. (right) Absorption and emission spectra of such micelle-isolated tubes. The correspondence between the peaks is nicely seen in the two spectra. From Ref. [4.13].

The problem with absorption and emission measurements on bulk samples is that we still have a huge number of different nanotuhes with varying diameter and chiral angle. Wc cannot tell from these measurements alone which absorption peaks correspond to two transitions of the same tube. To overcome this problem, O'Connell et n1.'4,131 measured the photoluminescence as a function of the energy of the exciting light, a so-called photoluminescence excitation spectrum (PLE). The photoluminescence of a particular tuhe, which is selected by a fixed detection energy, is high when the abqorption is strong for an excitation energy. Such a spectrum thus resembles the absorption of a single tuhe and allows the detcrrnination of the higher-lying van-Hove singularities. We can then try to use these energies for an assignment of the chiral indices nl and nz.

4.3 Photoluminescence Excitation - (n n z ) Assignment

'"

Bachilo et nl.L4 combined the information obtained from luminescence and lurninescenceexcitation spectra with that of resonant Raman scattering and attempted a full assignment of their micelle-covered tubes in solution. They measured the luminescence spectra as a function of excitation energy (PLE) and created a two-dimensional plot, see Fig. 4.7. The van-Hove singularities of many different nanotubes are nicely seen as dark spots, which, by the nature of PLE, relate absorption and emission of the same nanotube. This is the systematic advantage of PLE over regular absorption measurements, where a direct correlation between the two processes cannot be made and both absorption and emission spectra are much br~ader.[~"I

4 Optical Properties Figure 4.7: Two-dimensional plot of luminescence energy (x-axis) as function of luminescence excitation energy (yaxis). The darkcst spots denote the largest intensity; the black ellipse highlights the region of the E22 excitations and E l l emissions. Peaks in the lower part of the figure are belicved to come from E3? excitations; the line cutting through thc upper left corner is the exciting laser. After Ref. 14.161. Emission wavelength (nm) [c,+v,

transition]

The challenge is to assign systematically chiral indices to the intensity peaks in Fig. 4.7, which we plot on an energy scale in Fig. 4.8 (a). The assignment is still a difficult task as many different diameters and chiralities arc present in the sample. In Fig. 4.8 (b) we show in an analogous plot the rcsults of a third-order tight-binding calculation (Sect. 3.1.1) of the transition energies Ezz versus Ell.14 17] While they show a similar pattern to the experiment there are also some apparent deviations. Both plots show data points arranged in V shapes with the V opening towards higher energies and hence smaller diameters. The solid lincs forming the V correspond to moving in an armchair direction from a particular starting tube (vertical arrow in Fig. 4.9); the dashed lines connect tubes through an equivalent armchair direction (diagonal arrow). The points in the upper part in the calculation in Fig. 4.7 (b) belong to (nl - nz) mod 3 = -I nanotuhes, those below the V-rnininum to ( n l - n2) mod 3 = +l.

Figure 4.8: Plot of the cmission energy vprsus excitation energy (a) from the data in Fig. 4.7, and (b) from a third-order tight-binding c a l ~ u l a t i o n . [Solid ~ . ~ ~and ~ dashed lines connect tubes in two equivalent armchair directions, see Fig. 4.9. The similarity of the two patterns is apparent and may be used to perform an (nl,n2) assignment. Experimental data taken from Ref. [4.16].

4.3 Photolurnine.wmce Excitation - (nl, n z ) Assignment Figure 4.9: Possible chiral indices for nanotubes. The vertical and diagonal arrows correspond to two equivalent armchair directions as connected by solid and dashed lines, respectively, in Fig. 4.8. Tubes with gray hexagons were assigned to experimental features in Fig. 4.7. After Ref. [4.16].

Different values for EZ2in different tubes at the same Ellarise from the trigonal shape of the graphene Brillouin zone near the K point, which splits the energies of the mod 5 1 families, see sect. A difficulty in proceeding with the assignment is that the energies predicted by tightbinding are not known very well on an absolute energy scale, leaving too many possibilities for an assignment. However, if we knew the assignment of one of the tuhes, we would be able to derive that of the others according to the pattern identified in Fig. 4.8. Bachilo et 01. used the radial breathing mode (RBM) in Raman scattering to fix one particular nanotuhe; for the uncertainties involved in doing so, see Sect. 8.2. From the inverse-diameter dependence of the FBM frequency [Eq. (8.1)] with C1 = 223.5 cm-l and C2 = 12.5 cm-' they assign all 33 different peaks to specific nl ,n2 chiral nanotubes. On the basis of their assignment they found empirical rules for the diameter and chirality dependence of the first and second electronic transition in carbon nanotubes. Similar results were obtained by Lebedkin et ul.,14.201 who isolated and analyzed the emission from somewhat larger-diameter tubes produced from pulsed-laser evaporation. For diameters centered at 1.3 nm they found their luminescence and absorption energies to agree qualitatively with the results shown in Fig. 4.8 and arrived at similar conclusions. Given an assignment to chiral indices, the authors observed that the largest luminescence comes from nanotubes close in angle to the armchair direction. There are two possible explanations for the intensity dependence on chiral angle. The distribution of chiral angles is not homogeneous and most tubes have 8 close to the armchair direction. This explanation, which assumes equal quantum yield for all chiralities, contradicts, however, electon diffraction experiments on bundled tubes.14.211The isolation process might work better for armchair-like tuhes, although there is no apparent reason for this selectivity. The second possible explanation is that the quantum yield is different for different chiralities, i.e., the luminescence of a single tube with 8 = 0' is much smaller than the luminescence for one with 8 = 30". Although we do not know the physical mechanism for the difference in quantum yield, this explanation sounds more appealing than a selectivity in the growth or isolation process. The absence of zig-zag nanotubes may also be taken as an indication of an incorrect assignment of the anchoring tube via the RBM frequency. We note from Fig. 4.8 that the observed pattern is much broader, the V shape much more open than that of the calculation; also the V appears to broaden more to higher energies, i.e., to smaller diameters. If the pattern suggested by the lines in Fig. 4.8 is correct, this implies 3 . 1 . 1 . [ ~ . ~ ~ I 7 [ ~ . ~ ~ l

4

76

Optical Properties

that effects due to the curvature of the nanotube, not included in the tight-binding calculation, can be estimated from such a measurement. Ab-initin calculations, which are sensitive to curvature. should improve the understanding of the observed emission and absorption spectra. In Sect. 3.4 we saw that curvature effects are more pronounced for c h i d angles close to the zig-zag direction than for nearly armchair tubes. Since zig-zag tubes are at the end of the V lines, curvature explains the more open experimental pattern at least qualitatively. Surprisingly, in these experiments the ratio of energy of the second van-Hove singularity to that of the first came out to fi2/EI I = 1.8 when extrapolated to large diameters or zero energy. All theories neglecting electron-electron interaction predict a ratio EZ2/E11 = 2 in the large-diameter limit because the valence and conduction bands of graphene are linear at K (Chip 3). The so-called ratio problem could bc due to excitonic effects, i.e., a lowering of Ezz due to Coulomb attraction of excited electron and hole. It is actually independent of the precise chirality assignment, as long as the patterns are valid in principle, i.e., as long as the peaks indeed refer to first and second singularities in the band structure of carbon nanotubes. Deviations from a strict ratio = 2 are expected due to trigonal warping and, for small-diameter tubes, due to diameter-dependent curvature effects on the transition energies. As mentioned, however, the e~perirnents[~.~~1,1~.~'l indicate that also in the limit of large diameters the ratio does not approach 2, instead it remains at = 1.8, E22 #2

El 1

for d +

(ratio problem).

(4.11)

A factor of 2 is expected for the ratio in the limit of d -t = because graphite is a semimetal; its bands cross the Fermi surface in a linear fashion and from the quantization of the allowed k lines in nanotubes we get the factor of two. Kane and ~ e l e [ ~suggested . ~ ~ ] an interesting explanation for the ratio problem in carbon nanotubes. They proposc that because of the one-dimensionality of the carbon nanotuhe bands electron-hole interactions are particularly noticeable in the excited states. An electron-hole pair after absorption and relaxation to the

Figure 4.10: An explanation for the ratio problem (see text). (left) The absorption process (a) into the secund electronic hand EZ2 of a nanotube and subsequent emission (b) from Ell after relaxation. (middle)Dccay of an electron-hole pair from the minimum of the second excited band into hvo electronhole pairs in the first excited band (d,e). The energy is given in units of thc dimensionless parameter E = E d / 2 h v F . (right) The ratio EZ2/E11as a function of a coupling parameter a. The experimental range is indicated by the gray bar. From Ref. 14.221.

4.4 4-A-diometer Nanotuhes

77

bottom of the second band can decay into an electron-hole pair in the first excited state, see Fig. 4.10 (d). Electron and hole exchange linear (Ak) and angular (Am = 1) momentum in this process. At the same time another electron-hole pair is excited (e). The screening due to this relaxation process scales only the energy of the second band, since the lowest hand cannot decay into an even lower continuum and a ratio E22/E11 < 2 is obtained. Obviously the ratio depends on the coupling strength & for the decay process; in Fig. 4.10 the ratio E22/E11is plotted as a function of 8. The overlap with experiment is given in the gray region, thus & x: 0.2 can explain the experimental ratio problem. For details see Ref. 14.221. Summarizing, this new and exciting type of spectroscopy of carbon nanotubes has given the understanding of the band structure a significant push. Even though the chiral assignment is perhaps not final, it has become possible to directly investigate and identify many nanotubes with chirality-specific information. We expect much new insight into how physical properties of carbon nanotubes depend on chiral angle from such measurements.

4.4 4-A-diameter Nanotubes The optical properties of the small nanotuhcs that fit into the channels of zeolite have been calculated by various gro~ps.14.23]-[4 251 They are particularly useful for comparison with experiment, since firstly, their properties are probably most affected by the small radius, and, secondly, there are only three different chiralities expected, (3,3), (4,2) and (5,O); see also Sect. 8.2 for a discussion of the radial breathing mode detected in Rarnan scattering. Moreover, these small tubes are aligned in the channels, which allows an experimental separation of optical transitions with different polarization. Laslly, the depolarization effect can be verified in more detail (Sect. 4.1.1). In Fig. 4.11 we show the electronic band structure of the three tubes believed to grow in zeolite. They are remarkable in that they deviate quite significantly from what one would obtain from zone folding (see Sect. 3.2). In particular the zig-zag tube turns out to be metallic instead of semiconducting as its indices or a zone-folding calculalion would predict. This is due to the large rehybridization of a" and R* bands due to the strong curvature of the tube's surface, see Sect. 3.4. The metallic armchair tube (3,3), in spite of its similar radius, is less Figure 4.11: Electronic band structure of the (3,3), (4,2), and (5,O) nanotuhe as obtained from an ab-initin calculation. Note that the (5,0), prcdicted to be semiconducting from

zone Folding, actually comes out metallic. From Ref. [4.23].

(3.3)

(5.0)

(4.2)

4

4 3

2

2

2- '

8

5

1

0

0

-1

-I

-2

-2

-3

-3

4

r

$=I.z~

r

Ez0.74

r'

Wave vector kt f R 1 )

;4 2 8-4

78

4

Optical Properties

affected; note that the Fermi wave vector kF is significantly smaller than 2 ~ / 3 a(the value for large armchair tubes) and the next higher minimum lies at k > 2a/3a.[4.121,~4.231~[4~261-[4~281 Thc optical response can be calculated by finding the dipole transition matrix elements between occupied and unoccupied states and integrating them for the allowed transitions between the valence and conduction bands [Eq. (4.6)]. This gives the absorption of such a tube; a Kramers-hnig transform yields the real part of the dielectric function. We show in Fig. 4.12 the complex dielectric function for the three tubes in parallel polarization, obtained from the outlined procedure.[4.23]In the uppermost panel we plot the absorption as determined from EZ using the relation between the optical constants in the beginning of this chapter and an effective-medium average adequate for the filling fraction of tubes in the zeolite crystal. All perpendicular absorptions are expected to be much weaker than the parallel ones due to depolarization, see Sect. 4.1.1. For details we refer to Ref. [4.23]. Li ~t al.[4.121measured the optical spectra of nanotubes in zeolite, we show their spectra in Fig. 4.12. The spectra have a prominent absorption peak (A) at 1.37 eV that is nicely reproduced in the calculation. It corresponds to an optical transition at the point between bands of equal rn in the (5,O) tube as required by the selection rules. The transitions B and C are also well reproduced in energy, although somewhat smeared out in the experiment due to the rising plasmon background. The peak at 2.1 eV (B) is assigned to the (4,2) tube, the second, weaker contribution to the (5,0),and the peak at 2.9 eV stems from the (3,3) armchair tube. Note that all calculated energies are shifted slightly to lower energy (= 10 - 20%) due to the known LDA problem. These results also agree well with those published by Li et al.14 "1 and Liu and chamf426] A calculation in the time-dependent local-density approximation taking into account the depolarization effect in nanotubes is shown in Fig. 4.13. The absorption for an electric field perpendicular to the z-axis is displayed for both an isolated (3,3) nanotube and a crystal of nanotubes. This calculation confirms the hand-waving explanation of the nice polarized pho-

r

Figure 4.12: (upper panel) Calculated ahsorption of zeolite nanotubes (scale adjusted to the experiment). The panels on the right contain C. of a (3,3) (thin solid line), a (4,2) (dashed), and a (5,O) (thick solid line) tube for parallel polarization. From Ref. [4.23]. (lower left) Optical spectra for various angles between E and the z-axis. 0" corresponds t z 1) E . From Ref. [4.12]. 05

1.0

1.5

2.0

2.5

3.0 3.5

motm Energy (sV)

4.0

u

4

Energy (eV)

4.5 Bundles ofNanotubes 4.13: Optical absorption calculated by timedependent density-functional theory for a ( 3 3 carbon nanotube. Uppcr and lower panel on the left correspond to perpendicular absorption in isolated tubes (thin lines) and a solid of (33) tubes (thick). Dashed and solid lines are witbout and with local-field effects. On the right parallel absorption in the three zeolite-sized tubes is shown. After Ref. 14.251. Figure

1

2

3

4

5

6

7

energy (eV)

energy (eV)

tograph in Fig. 4.5. The depolarization effect is nicely seen in the difference between dashed and solid curves, it is particularly strong in isolated nanotubes. In parallel polarization the absorption spectra and their assignment to optical transitions are the same as described by Machhn et al. above.

4.5 Bundles of Nanotubes For a long time, before isolated tubes in solution became available (Sect. 4.2), the only possibility to study the optical properties of nanotubes was to measure a sample with bundles of tube^.^^.^^]-[^."] In the analysis, assumptions about the distribution of tubes could be tested against the experiment. Because the absorption spectra are generally broad and unstructured and usually have a strong, linear background due to strong higher excitations, the information obtained is, however, not very specific. This background has two contributions. It stems partly from the higher-lying plasmon excitations in single-walled nanotubes, which appears in isolated tube spectra as The second contribution and the strong broadening of the absorption peaks is due to the bundling of the tubes. In bundles, the tube-tube interaction introduces an electronic dispersion perpendicular to the k, axis, see Sect. 3.5. Therefore, even for a bundle composed of exclusively one type of tube we expect broadened and unstructured optical absorption spectra.14.281 Nevertheless, transitions corresponding to the first and second singularity of semiconducting nanotubes and to the first singularity o f metallic nanotubes can be resolved from experiment, see Fig. 4.14 for an example. In an attempt to analyze the optical spectra of bundled nanotubes the linear background is usually subtracted. A spectrum as in Fig. 4.14 is then obtained. To extract, e.g., a diameter and chirality distribution these spectra are then fitted with the E l 1 and E22 transition energies of isolated nanotubes in a tight-binding model. Although an estimate of the diameter can surely be found by this procedure, detailed conclusions about diameter andlor chirality distributions should be treated with care. Experimental and theoretical work on the bundle dispersion, however, remains a challenge for the future.

4

Figure 4A4: Optical ahsorption from a sample with bundlcd nanotubes after background subtraction (solid line). The dotted line is a fit to a distribution of nanotubes with all chiralities included. The ratio problem, i.e., EZ2 < 2E11is seen on the right (for a discussion, scc Sect. 4.3 and 4.6). After Ref. 14.341.

Optical Properties

-

-E;a z

5

,g

0"

4.6 Excited-state Carrier Dynamics From the absence of strong luminescence in nanotube bundles we suspected that probably rapid transfer processes take placc from semiconducting to metallic nanotubes in bundles of tubes. In the mctallic tubes carriers can relax to the Fermi surface and recombine nonradiativcly. Measurements of the relaxation time in metallic tubes have been performed by Hertel and ~ o o s . [ They ~ ' ~ used ~ time-resolved photoemission to probe the distribution and lifetime of carriers near thc Fermi surfacc. In Fig. 4.15 we show schematically the Fermi surface at equilibrium and at high tempcratures, excited by a laser beam in the experiment. Clearly seen in the experimental photoemission spectrum in the lower panel is the temperature increase in the distribution of the carriers near the Fermi surface. By following the photoemission as a function of time the authors were able to determine a characteristic time of about 200fs for the carriers to obtain a didribution that they could fit with a Ferrni-Dirac function. Hence excited carriers in metallic tubes lose their energy rapidly, in less than 200 fs, and it is not surprising that they quench the lurnincscence of nanotuhe bundles efficiently. Using optical time-resolved spectroscopy one can address the lifetime of carriers not only near the Fermi level hut also at higher energies in the van-Hove singularities. Tbo-color pumpand-pmbe experiments are the method of choice hcre. Briefly, in a two-pulse experiment, the first laser pulse excites the system, say, into the second conduction suhband. The second h e r Figure 4.15: (upper panel) A Fermi distribution of carriers near the Fcrmi level at room temperature (dashed line) and a1 a higher, laser-heated temperature. (lower) The experimentally determined carrier distribution about 1 ps after excitation is vlotted as difference to the room-temperature distribution. After Ref. [4.35].

C

- mrnternwrature

g

2 X

---

Fnrmi-Dime fit

@ T=W5K Experiment

E=. low-energy cutoff J U)

X

\

rl

4.6 Excited-state Carrier Dynamics

81

pulse, in general of a different photon energy, can probe the population or depopulation of the same or of a different state, say, the first excited band. The physical mechanism behind the probe is that the absorption of a system depends on the population of the ground and excited states, Since the population was changed by the pump pulse, we can follow the relaxation of the carriers in time by looking at the transmission of a sample. Such experiments were performed by Lauret et a1.14.361 and Ichida et aL[4.371on bundled nanotubes. Figure 4.16 shows the change in transmission AT of bundles when exciting into the first (black line) and second (gray) van-Hove singularity of semiconducting In the insct we display the optical absorption spectrum of bundled tubes; the arrows point at the pump-and-probe energies in the two experiments. Obviously, the relaxation time is much shorter in the second lhan in the first excited state. Lauret el nl.r4.361 obtained a carrier lifetime oS 1 ps in the lowest valence and conduction band. In the second van-Hove singularity the lifetime was an order of magnitude shorter, around 130 fs. Free carriers excited into the second valence and conduction band thus rclax very rapidly to the band gap of the tube. From there they tunnel either into metallic nanotubes or into semiconducting tubes with a smaller band gap. In metallic nanotubes the photoexcited carriers recombine nm-radiatively; in small band gap (= 0.2eV) semiconducting nanotubes, phonon emission can be another relaxation mechanism. The lumineqcence from isolated carbon nanotubes is expected to have a different behavior. The rapid relaxation channcl to metallic tubes is not present because the tubes are not bundled, and while the metallic tubes relax just the same as in bundles, the semiconducting ones show a strong luminesccnce, see Sect. 4.2. This luminescence was found to have a much longer lifetime than any of the measurements in bundled tubes. In Fig. 4.17 (a) we show the decay of the luminesccnce as determined from time-resolved picosecond photoluminescence spectroscopy. The luminescence corresponding to thc E l 1 transition of an (8,6) nanotube has a recombination time of 30ps. This is one ordcr of magnitude longer than the band gap relaxation in bundled tubes and explains very well the absence of luminescence in nanotube bundles. From this lifetime we also learn that the intrinsic width of the luminescence of a single tube muqt be even smaller (< 0.1 meV) than observed, e.g., in the cw (continuous wave) experiments in Fig. 4.7. lndeed narrower luminescence lincs (z 10meV) were observed in experiments on unprocessed isolated n a n ~ t u b e s . l ~Figure . ~ ~ ] 4.1 1 summarizes the carrier dynamics as observed experimentally. The figure shows the band structure of an (X,6) nanotube Figure 4.16: Normalized change of the transmission of bundled nanotubes for pump and probe energies both at 0.8eV (black line) and 1.47 eV (gray line). The inset shows the absorption spectrum of nanotube bundles. The arrows indicate the cnergies used in the pump-and-probe experiments, From Ref. [4.361.

0

I Time delay (ps)

2

82

4

Time f w d

Optical Properties

Wave vector (I o

- 1\~ ')

Figure 4.17: Photoexcited carrier dynamics in single-walled carbon nanotubes. (a) Time-resolved photoluminescence measurement on an isolated (8,6) nanotube. Gray dots are the raw data; the line is a fitted convolution of the system response (40 ps) and the luminescence signal. (b) Band structure of an (X,h) nanotube in the third-order tight-binding approximation in a single-particle picture. The arrows indicate the carrier dynamics in isolated nanotubes (relaxation and recombination)and in nanotuhe bundles (relaxation and tunneling into another nanotube). (c) Lower trace: energy gained by the relaxation of electron and hole from the second into the first subband in the (8,6) tube. Upper trace: Energy needed for an excitation of an electron from the ground into the excited state. The dashed lines denote the corrcctions by taking into account electron-hole interaction as described in the text. Note the expanded energy scale in (c).

calculated with the tight-binding approximation.[4.171An incoming photon with energy E22 excites an electron into the second van-Hove singularity. The carriers relax to the bottom of the first valence and conduction bands on a very short time scale. In isolated nanotubes the electron and hole recombine across the band gap after 30 ps. In contrast, in bundled nanotubes the much faster relaxation channel is tunneling into nearby metallic tubes (1 ps). In Sect. 4.3 we discussed electron-hole interaction and the formation of electron-hole pairs in carbon nanotubes by Coulomb interaction. We found that the optical transition energies are altered fundamentally in the limit of large-diameter tube^.^^.'^] For the (8,6) nanotube in Fig. 4.17 (h), however, we calculated the electronic dispersion neglecting Coulomb interaction. The important process for solving the ratio problem in large-diameter nanotubes was the decay of an electron-hole pair in the second subband into two pairs in the first excited state. Let us now estimate whether this process is possible for the small-diameter (8,6) nanotube (d = 0.9nm). In Fig. 4.17 (c) we show the electronic energy gained by the simultaneous relaxation of an electron and a hole from the second into the first subband (full black line); the gray-shaded area shows the energy that is necessary to excite an electron from the ground state into the first excited state. The process suggested hy Kane and Mele1422] is possible, if the two curves show an overlap, which they do not. In the single-electron picture the energy gained by the relaxation is thus not sufficient for the excitation of another particle. The band structure in Fig. 4.17 (b) that we used for the calculation of the energies in (c), however, was calculated neglecting electron-hole interaction. Including Coulomb interaction might strongly shift the electronic states to different energies and introduce an overlap between the two curves in Fig. 4.17 (c). We estimate the exciton binding energy for the first excitonic

4.7 Summury

83

state by the well-known relationr4,113r4.381

where m is the electron mass, E, = 2.1 is the dielectric constant of the tube as estimated by 221 Kane and ~ e l e , L ~Ry=13.6eV is Rydberg's constant, and y is the reduced mass of the exciton defined by the masses of the electron me and the hole mh

The electron and the hole are not fully symmetric in the third-order tight-binding approximation, because the electronic band structure of graphene is slightly different below and above the Fermi energy. For an order-of-magnitude estimate for the exciton binding energy, however, me = mh is a sufficiently good approximation. The electron mass in the first mel and second me*conduction band are given by

Fitting the band structure of the (8,6) nanotube in Fig. 4.17(b) with parabolas around the m~. the electron masses minima we obtain me, = 6.8 and me:!= 4.8 x 1 0 - ' ~ e ~ s ~ / Inserting into Eq. (4.12) we find a binding energy of20 meV and 12 meV for the first and second excited state, respectively. This correction to the tight-binding energies is shown in Fig. 4.17 (c) by the dashed lines and is only a small correction to the electronic energies calculated without Coulomb interaction. For smaller tubes like the (8,6) (d = 0.911,) where the approximation of linear graphene bands is no longer valid, the interesting effect of Kane and Mele thus becomes lew likely. Nevertheless, in large tubes it should be observable.

4.7

Summary

In summary, optical experiments have become a useful method for the investigation of nanotubes, Oriented tubes allowed a detailed comparison of absorption and emission properties predicted by calculations of the electronic band structure. We saw the effect of the selection rules including the depolarization due to the cylindrical shape of the nanotubes and presented a new improved Kataura plot. Much progress was made when it became possible to keep nanotubes from rebundling; the long-missed luminescence appeared quite strongly. In combined photoexcitation luminescence and Raman experiments the first serious attempt to assign chiralities to spectroscopic features was made. Finally, we studied the dynamics of carriers in the first and second excited states. Carriers in debundled nanotubes have an order of magnitude longer lifetime in the first excited state than in bundles, making isolated nanotubes a promising candidate for light cmitters.

Electronic Transport

Carbon nanotubcs can be metallic or semiconducting depending on their diameter and chiral angle. Metallic tubes, in particular, the truly metallic armchair tubes, can act as tiny wires. The one-dimensional nature of thcse nanowires gives rise to a variety of exotic phenonema like Coulomb blockade, Luttinger-liquid behavior, or proximity-induced superconductivity. Semiconducting nanotubes, on the other hand, can be envisioned to act as transistors in the nanoworld. So far, however, even the most basic requirements for possible applications as, for example, separating metallic and semiconducting nanotubes remain challenging, although some progress was made on this topic.15 l1 This chapter will concentrate on the physical phenomena associated with transport in single-walled carbon nanotubes. It will remain to be seen whether the topics dicussed hcre are just exciting or will lead to useful applications. In the following we first givc an overview d the expected transport properties in ideal nanotubcs and explain how to perfom transport measurements on isolated nanotubes or small bundles of tubes. We then discuss the Schottky barriers that form at the junction between a tube and a metal. Section 5.2 deals with the scattering of electrons in single-walled nanotubes. We show that armchair nanotubes arc ballistic conductors because elastic scattering is suppressed by the symmetry of the conducting electrons and because the local potential of defects is small. Electron-phonon interaction, in particular with optical phonons, seems to be the main scattering mechanism. The last two sections turn to more exotic effects jn electron transport in nanotubes. We first discuss the Coulomb blockade and then briefly turn to Luttinger-liquid behavior in single-walled carbon nanotubes.

5.1 Room-temperature Conductance of Nanotubes In an armchair nanotube - like in graphene - the valence and the conduction band cross at the Fermi level. The tube has two bands that can carry currents close to EF, or two conducting channels. In each band we can put two electrons with opposite spin. Thus, an ideal armchair nanotube with perfect contacts to a metal has a conductance of 2Co = 4 e 2 / h , where Go = 75 pS is the conductance quantum including spin degeneracy. Figure 5.1 (a) shows the conductance (full line) and (b) the band structure of a (5,S) a m chair nanotube; both were calculated from first principlcs.[5~2]If the Fermi level is shifted, e.g., to AEF = -0.2eV [upper dashed line in (b)] only the two crossing bands with m = n = 5 conduct the current as discussed above. At even larger AEF = -1.8eV (lower dashed line) we find four k points in the valence bands intersecting the Fermi level. Since the lower-lying valence bands are two-fold degenerate, the conductance is now 6Go. The conductance is reduced by Go at AEF = -2cV, because one of the rn = 5 bands then is empty as can be seen in

I

86

5

Energy (eV)

r

nla

Electronic Transport

r

nla

Wave Lrector kz

Figure 5.1: (a) Conductance of a (53) armchair (full line) and a (9,O) zig-zag nanotube (dashed line). The conductance is quantized in multiples of Go. While the truly metallic armchair nanotube has a finite conductance at the Fermi level, the conductance of the (9,O) zig-zag tube drops to zero at the Fermi level. The energy scale is such that the Ferrni level is at 0. (b) and (c) Rand structure of (h) a (53) and (c) a (9,O) nanotube. The dashed lines represent the Fermi level shifted to AEF = -0.2 and - 1 .Rev. See also Chap. 3.

Fig. 5.1 (b). The dashed lines in Fig. 5.1 (a) are for a quasi-metallic (9,O)zig-zag nanotube; its band structure is shown in Fig. 5.1 (c). Note the secondary gap at EF, which comes from the curvature of the nanotube wall, see Sect. 3.4. At very low temperature this gap prevents transport in non-armchair ( n l - n 2 ) / 3 integer nanotubes. In a zig-zag tube all bands are two-fold degenerate close to EF and the critical point is the point. At AEF = -0.2eV one degenerate band gives rise to a conductance of 2Go; at - 1.13eV we find three bands, i.e., C = 6 Go. Finally, semiconducting nanotubes have band gaps between = 0.5 - I eV, except for thc large band gap they behave exactly as the quasi-metallic (9,O) zigzag tube in Fig. 5.1 (a) and (c). To measure electrical transport in single-walled carbon nanotubes the tubes are usually dispersed or CVD-grown on a prepatterned substrate. The substrate is a silicon wafer (used as the back gate in the experiments) with a 100 - 1000-nm thick SiOz layer. On this layer a large array of metal electrodes is prepared by elcctmn-beam lithography. After depositing the tubes, the sample is scanned with an atomic force microscope to find tubes that - by coincidence bridge two or more electrodes. The diameter of the nanotube or the small bundle of nanotubes is determined by their apparent height under the AFM. After a likely candidate for transport measurements is identified the transport characteristic is measured to see whether there is any contact at all between the electrodes and the tube, and, given this is the case, whether the tube is metallic or semiconducting. With the procedure described, many interesting physical phenoma were investigated, as we discuss in the next section. It was also demonstrated that carbon nanotubcs can be built into transistors and logic elements.[531-[5.61 However, it is quite obvious that this approach is not suitable for application in technology. One of the biggest problems is that it is not possible to separate metallic and semiconducting nanotubes before depositing them, let alone to grow tubes to design. A solution to the separation problem was suggested by Krupke st u Z . ~ ~ . ' Figure 5.2 shows two different types of behavior of the room-temperature conductance (zero bias) v m u s gate voltage. In Fig. 5.2 (a) the conductance is independent of the gate volt-

r

Figure 5.2: Conductance of el,,l . (a) a metallic and (b) a scn~iconducting nanotubc at rooni 30 tempcraturc. Thc Insets show .ichematlcally ihe electronic ;n" 2- 02 hand structure of the tubes near the Fermi level. From Ref. [5.7]. 4

(a)

15

"'L,

-

i

r; cw

,

;

-4

4

I/;

o (v)

4

age, a behavior attributed to metallic nanotubes. The crossing of the valence and conduction band in the metallic tubes provides conducting electrons independently of the gate voltage. This is schematically shown by the inset in Fig. 5.2 (a). The conductance in semiconducting nanotubes, in contrast, changes by orders of magnitude when changing V,, see Fig. 5.2 (b). In semiconducting tubes an asymmetry between the turn-on gate voltage below and above the Fermi level was observed. The tubes became conducting for negative V,, at a lower ahsolute value than for positivc V,, which was naturally attrihuted to an intrinsic p-type doping of the however, showed that the asymmetry between the two cont~bc.["~]-["] Heinze et a1.,[~~"'1 ducting channels is not due to a doping of the lube, but rather due to Schottky barriers that form at the tuhe/mctal interrace, which we discuss now. Figure 5.3 (a) shows the calculated conductance of a semiconducting nanotube which is connected by Schottky barriers to a metallic lead.L5 "'1 The different curves are for different work functions of the metal. Changing the work function can be achieved by exposing the

I

-0.8

-0.4 0.0 0.4 Gate Voltage (V)

0.8

10 Gate Voltage (V)

-10

0

I

I

I

I

I

0 2 4 6 8 1 0 Distance from Junction (nm)

Figure 5.3: Schottky-harriertransistor with semiconductingnanotubes. (a) Calculated conductance vcrsus gale voltage for different work functions of the metal with respect to the nanotube work function. Full (dashed) lincs awurncd a larger (smaller)work function for the metal than for the nanotube; the difference is highcr for darker gray. (h) Measured device characterixtic for diCferent work functions of the metallic clcctrodes. The work function was changcd by exposing the device to oxygen. (c) Schematic band struclure of the nanotube close to thc mctal-nanotubejunction. Full lines: zcro gate and hias voltage, the work functions of the metal and the tuhe differ by 0.2 eV. Dashed lines: Vp = -S00mV, Vb = 0. From Ref. [5.10]; see there for details.

device to oxygen, which previously was belicvcd to dope the nanotubes. In the figure we see that the conductance then decreases for one sign of the gatc voltage, whereas it increases for the other. Thc range of gate voltage where the device is non-conducting, in contrast, does not vary. Experimcntiilly the same behavior can be seen in Fig. 5.3 (b). For a doping of the tube we would expect a shift of the minimum of the curves to a different gatc voltage, while their shape would remain unchanged for moderate doping concentrations. In Fig. 5.3 (c) we show the hand structure of a nanotube closc to the metal junction (full lines); the work function of the metal was assumed to be larger than that of the tube. Far away from the junction the Fcrmi level of the nanotube is in the middlc between the valence and the conduction band. Applying a positive or negative gate voltage, we can switch the nanotube into the conducting state. At negative voltage (dashed lines) the Schottky barrier is thinner than for a positive gate voltagc. Consequently, a larger current flows through the harrier for the p channel than for the n channel at the same absolute Vg. Owing to the Schottky barriers at the metalltube interface, the operation of nanotubes field-effect transistors ih not based on modulating the . ~an~ . ~ ~ ~ charge density in the channel, but rather on a modulation of the contact b a ~ ~ i e r sAs important consequence, the contact geometry influences the conductancc because the electric field at the interface determines thc width of the Schottky barrier. Lct us return to metallic nanotubes, ie., t u b ~ sshowing a conductance as in Fig. 5.2 (a). As we saw above, these tubes wcrc cxpected to have a conductance of 2Go or a resistance of 6.5 ka.Looking at Fig. 5.2 (a) we find, however, the measured conductance to be much lower than the expected one. The first transport measurements reported a two-tcrminal resistance on the order of M a , i . ~ .most , of the applicd voltage dropped across the barricrs bctween the tube and the contact and tunneling dominates the electrical transport. The origin of thc highcontact resistance remains an open qucstion. It might be related to a combination of extrinsic - like the quality of the electrodes - and intrinsic factors. For example, ~ersoff[" '1 suggested that tunneling into an armchair nanotube from a gold contact is suppressed by the nanotube's quantized electronic states. Nevertheless, highly transparcnt contacts were reported for metallic nanotubes with a conductance approaching the theoretical limit of 2 ~ ~ . l ~ . ~ ~ 1 , 1 ~ In theqe devices the conductance oscillated as a function of gate voltage, which was attributed to Fabry-Perot interferences along the nanotube.

5.2 Electron Scattering With highly transparcnt contacts between the nanotube and the electrode resistances close to the ideal limit of one fourth of the quantum resistance were observed e ~ ~ e r i m c n t a l l ~ . l ~ ~ ~ ~ 1 ~ ~ Onc might wonder why the conductance in a real tube is so close to the theoretically cxpccted limit. Real nanotubes do not have the perfect structure wc assume theoretically; they have defects that should scatter electrons. Electron-phonon interaction leads to inelastic scattering in s~lids.["~]-["'~"lc discuss the scattering of electrons for thc example of armchair nanotubes. Similar arguments hold for quasi-metallic tube^.^^^"^ There are two arguments why elastic scattering is suppressed for the electrons ncar the Fcnni level in metallic nanotubes. The first one, put forward by White and ~odorov,["l~1 is nicely understood in a hand-waving picture. The electronic wave functions of a tube closc to the Fermi lcvcl typically have a shape as shown in Fig. 5.4. The electronic states are smeared

5.2 Electron Scattering

89

Figure 5.4: Electronic wave function of a (1 0,10) armchair nanolube close io ihe Permi level. Light and dark gray indicates the sign of the wave function. This is the eigenstate that transforms even under the vertical mirror plane.

around the nanotube's circumference. If some disorder, e.g., an impurity, is present in the wall, the electron will feel only an impurity potential averaged over the circumference. The larger the diameter of the tube the smaller is the influence of the impurity. In single-walled carbon nanotubes of reasonable size d z I nm the electrons at the Fermi level can thus travel without scattering.["'7] The limit of this picture is set for diameters wherc the higher-lying electronic states come close in energy to the conducting channels. We then obtain a finite coupling, i.e., scattering, between the crosqing valence and conduction m = n bands and bands with other rn quantum numbers. Thus, for very large-diameter nanot~ibesballistic transport along the axis breaks down. The second argument is based on symmetry. Although the following symmetry discussion is strictly valid only for armchair nanotubes, it turns out that a similar supression of backscattering occurs in zig-zag and chiral quasi-metallic t u b e ~ . L ~ . ' '"~I ] , [The ~ wave functions belonging to the two conducting channels in an armchair nanotube are orthogonal to each other, because they have different parity under the vertical reflection as we saw in Sect, 3.4.1. As long as the mirror symmetry of the nanotube is not severely damaged by the impurity, a coupling of the two conducting electrons is forbidden. Figure 5.5 (a) shows ah-initio calculations of the conductance of an armchair nanotube with a pentagon-heptagon defect.15.21%15.201 In Fig. 5.5 (b) the atomic structure of an armchair tube with such a defect is displayed. Obviously, the tube still has the horizontal and one vertical mirror plane. Thc two bands crossing at the Fcnni Icvcl, therefore, cannot mix. The calculated conductance for the (10,lO) nanotube in Fig. 5.5 (a) is indeed not affected around EF. Further away from the Fermi level there arc two pronounced dips in the conductance. Note, however, that G drops only to 2p2/h and not to zero. The decrease in conductivity comes from a reflection of the n or 7 ~ *electron by the impurity states. Since the pentagon-heptagon-derived states also transform as even or odd under the mirror planes, an impurity statc can only reflect one of the conducting electrons for a given energy. Choi et a1.15,201 ciilculated the conductance of an armchair nanotube with boron and nitrogcn impurities. Again the conductance at the Fermi level remained the same as for the perfect tube. In contrast to the pentagon-heptagon dcfect discussed above the B and N impurities break the vertical mirror symmetry. They preserve, however, the horizontal mirror plane, A state with a wave vector k, along the nanotuhcs axis and a quantum number of m = n as for the two electrons close to EF can - in the presence of the horizontal mirror plane - only be reflected into -k, and rn = -n = n, i.e., exactly the same states with a reversed k , wave vector. The last equality ariscs bccause the rn quantum number is only defined modulo 2n, scc Chap. 2. Therefore, as for the 5-7 defect, for elastic processes an electron cannot interact

5 Electronic Transport

Figure 5.5: Conductance of an armchair nanotube with a 5-7 defect. (a) The broken line shows the conducmce o i a perfect (1 0,10) nanotubc. Thc dashed line is for the nanotube with a pentagon-heptagon defect. From ReC. 15.201. (b) Structure of a (5,5) nanotube with a pentagon-heptagon defect. The atoms making up the defects are shown in black.

with electrons in the second conducting subband. As a consequence the conductivity close to El; is quite robust with respect to impurities.["18]~[5.19]-[5~21] This situation only changes when the potential associated with the impurity is short compared to the carbon-carbon distance as pointed out by Ando and ~akanishi.L' An example of such a short-ranged potential is a showed that indeed the vacancy in a nanotube. The ab-initio calculations by Choi et al.[5.201 conductivity around the Fermi level is lowered for this kind of defects. A direct experimental vcrficiation of these ideas is quite challenging. One could, r.g., consider introducing defects into a highly conducting nanotube and lhcn measuring the transport properties again to see whether they were altered, The two conclusions we can draw from existing transport measurements are that some tubes reach the ballistic transport limit and that in semiconducting tubes backscattering is present.r"ll Since the conducting electrons in semiconducting tubes (under large applied gate voltage) come from degenerate states, i.e., the electrons have the same symmetry, backscattering is allowed here in contrast to metallic tubes. McEuen et a ~ . [ ~ estimated .*~' a mean free path on the order of 100 nm from low-temperature transport measurements on semiconducting nanotubes. This is in good agrecment with the report by Javey et a/.,r51' who found that the transport in semiconducting tubes approaches the quantum limits for tuhes of 300 nm in length. In very long nanotubes (3 pm), however, the conductance dropped by a factor of 20, which we further discuss in connection wilh electron-phonon scattering. Crystal imperfections and other electrons are not the only source of electron scattering. In many cases the interaction with acoustic or optical phonons is a dominant relaxation process. Let us first consider which phonons are required for a coupling between the Ferrni electrons in armchair nanotubes. Figure 5.6 shows schemalically the valence and conduction band at EF in an armchair nanotube. A non-equilibrium electron distribution was achieved by applying an electric field. In the low-energy regime electron-phonon scattering requires modes with small but finite wave vectors to scatter an electron from subband 1 into subband 2 via process (a). This is a quasi-elastic process, because the energy of the acoustic phonons is very small. It changes the momentum of an electron, but to first order preserves its energy. Process (b)

'"

Figure 5.6: Scatteringby phonons in arrnchairnanotubes. The electronic band structure of an armchair nanotubc is shown by the two straight lines.

The gray lines indicate the non-cquilibrium electron population because the electrons were accelerated in an electric field. An electron in the first ( I ) subband can scattcr Into the second subband (2) by (a) ernitt~ng (or absorbing) acoustic phonons with a small wave veclor q. Process (h) require\ phonom wiih high energy like optical phonons or large-q acoushc phonons. In the latter case thc scattering takes place between kF and -kp.

in Fig. 5.6 can be mediated by optical phonons with q % 0 and q = 4~131.2or by acoustic phonons with q = 4 ~ / 3 u .The large phonon wave vcctors correspond to scattering from the Fermi point at 2 ~ 1 to3the ~ one at - 2 ~ / 3 a ;these two points are a single point in thc reducedzone scheme. Note the close similarity of electron-phonon scattering in electronic transport to the double-resonant Raman process, which is discussed in Chaps. 7 and 8. Scattering an electron Srom the first into the second subband requires a phonon that reverses the vertical mirror parity while conserving thc angular momentum quantum number in. This condition is fulfilled by A2, = B; or A , , = B i phonons. In armchair nanotubes there arc three phonons belonging to these symmetries, the twiston, the radial optical, and the axial optical phonon. The latter two are optical modes with an energy c)f 100 - 200 meV. At low energy the accelerated electrons do not have sufficient energy to cmit an optical phonon. The twiston is an acoustic tnode that is known to open a gap at the Fcrrni level in armchair nanotubes.L5.221-15.251 It was predicted to yield a resistivity that increases linearly with temperature.L5241 For the very small q vectors required for process (a), the electron-twiqton coupling is weak since at q = 0 thc twiston is simply a rotation about the nanotube's axis. This movemcnl does not affect the electrons of a nanotube.[' 22],[5

''

Further application of the electric field accelerates electrons to energies whcre process (b) in Fig, 5.6 bccomes possible, i.e., the emission of high-energy phonons with very small or 4 x 1 3 wave ~ vcctors. The axial high-energy phonon is known to couple particularly strongly It belongs, moreover, exactly to the B i to clcctrons in nanotubes and graphitc.["7],[5 1' reprcscntation required for proceqs (b). We thus expect that electrons with energies around 200 mcV above the unoccupied statcs arc efficiently scattered by electron-optical-phonon interaction. The energy of the radial optical and the twiston modc at large q are not known from experiment; theoretically thesc modes are usually found in thc range 100 - 150 me^.^^.^^^-[^."^ The relativc magnitude of the electron-phonon coupling of the axial optical and the two other phonons has not yet been considered. Comparison of an ah-initio calculation of the axial optical mode with a tight-binding calculation of the twiston mode suggests that electron-phonon coupling is an order of magnitude weaker for the t w i s t ~ n . [ " ~271 ~ ] Given ~ [ ~ the large differences between the two models this comparison is, however, only a rough estimate. From the above discussion we conclude that electron-phonon scattering plays only a marginal role if the energy difference between the occupied and unoccupied statcs in the non-equilibrium cncrgy distribution is less than 100 - 200meV. Once an electron reaches the threshold energy it can, however, relax efficiently by emitting optical and zonc-boundary phonons. A steady-state population is thcn established like the one in Fig. 5.6. Yao pi a1.15.321

5

Electronic Transport

Figure 5.7: (a) Current I in a nano1uhe as a function of the applied bias voltage Vb for three different temperatures. Thc inset shows the resistance R = V b / l of the sample. (b) Calculated I - Vb charactcristic assuming elastic scattcring and scattering by optical phonons. From Ref. [5.32].

showed that under these assumptions the limiting current for a nanotube is given by

EPh = 100 - 200meV is the phonon energy. In Fig. 5.7 (a) we show current-voltage characteristicq for bias voltages up tc) 5 eV. The measurements confirm that carbon nanotubes can indeed carry currents in the p A range; moreover, the current starts to saturate for high bias voltages. From the inset of the figure wc find that the resistance increases linearly with the applied bias voltage. This is readily understood in our simple picture of electron-phonon scattering. The higher the voltage the more rapidly electrons reach the threshold energy and are scattered by phonons. Yao et estimated the mean free path l,,, of an electron before it is scattered. The change in the momentum k of an electron in an applied electric field E is

with the linear electronic dispersion E* = hvF. k this yields

and, finally, the resistmnce due to phonon scattering is simply the ballistic resistance of the two conducting channels timcs the ratio between the nanotubc length L and the mean rree path lph.["'] In Fig. 5.7 (a) it can be seen that at zero bias voltage the resistance (around 40kSZ) is larger than onc fourth of the resistance quantum expected for ballistic transport. To account for this we introduce a finite elastic scattering term with a voltage-independent mean free path of I,. Then the total resistance is

The resistance has a linear dependence on the applied electric field or the bias vollagc in agreement with experimental findings. Following essentially the ideas outlined in the last

5.3 Coulomb Blockade Figure 5.8: Dependence of the mean free path for elast~cscattering I, on nanotube diameter. Thc values were calculated from the measured conductivity In wm~conductingnanotubes of = 300nm length rcportcd in ReC 15.61 and assuming a resistance as givcn In Eq. (5.4). The line is a linear fit to the data w ~ t hthe pomt at 3.5nm omitted from the regrcsGon.

Tube diameter (nm)

paragraph, hut with a refincd analysis, Yao ~t a/.15.321 calculated the I - V, characteristic in Fig. 5.7 (b), which matches the measured curve excellently. Equation (5.4) implies some general conclusions about the saturation current and the dependence of the resistivity on nanotube length, which wc can test against experiments. Javey et ul.IS reported nearly ballistic transport in semiconducting nanotubes with Pd contacts. They performed various transporl measurements for tubes with difrerent diameters and lengths and found that the conductancc of the nanotubes depends strongly on these two geometric parameters. Here the gate bias at which semiconducting nanotubes becomes conductive corresponds to the conductance at zero bias in metallic nanotubes. On the other hand, the saturation current was independent of tube length and reached 25 pA as for metallic t u h e ~ . [ ~ . ~ * ] The latter finding is in very good agreement with Eq. (5.1) which states that I[,,, due to phonon scattering is 5olely given by the phonon energy. The bias voltages necessary to obtain a similar resistance in a short (300 nm) and a very long (3 pm) nanotube with the same diameter differed by an order of magnitude (on the order of 100 mV and I V for thc 300-nm the 3-pm long tube, respectively). This is casily understood by looking at Eq. (5.4) and assuming I , and lpElto be the same for nanotubes of different Icnglh. The mean free path of electrons lor elastic scattering can bc estimated from the conductancc after turn on. From the values rcported by Javcy et we find that I , is of similar order for long and short nanotubes. The much larger resistance in longer nanotubes is qimply duc to their length as given in Eq. (5.4). Finally, an interesting point arises when we plot the mean rree path for elastic scattering as a function of nanotube diameter as in Fig. 5.8. Most of the points clearly follow a linear dependence; the fit extrapolates to zero for vanishing diameter with a slope of 40. In the beginning of this section we introduced the idea of White and ~odorovl"'~1that scattering in nanotubes decreases with increasing nanotube diameter. Although this argument waq originally developed for metallic nanotubes, the mean free path in semiconducting nanotubes secms to show a similar trend.

"

5.3 Coulomb Blockade Coulomb blockade is a quantum effect in t u n n c ~ i n g . [ ~R41~ ~TO ~ ]explain ~ [ ~ thc basic idea we use a quantum dot with equidistant energy levels instead of a nanotube, Consider the situation in Fig. 5.9 (a). The quantum dot is coupled by tunneling barriers to two metals. A finite bias 4 is applied to shift the Fermi level of the metal to higher energy e $ = e 2 / 2 c d with respect to the quantum dots, where Cd is the capacitance of thc dot. The small black circlc to the left

5 Electronic Transport

Figure 5.9: Schcrnatic tunneling through a quantum dot with tunnel barriers to two met&.. (a) The first four levels of the dot are occupicd by electrons. The first unoccupied lcvcl is e 2 / 2 c below the Femi cncrgy of the metal. (b) An electron tunnclcd from the metal into the quantum dot. Bccause of the Coulomb energy the potential energy of the quantum dot increases hy e2/c.(c) The electron is at the othcr s ~ d of c the dot and the energetic situation as in (a) is restored.

represents an electron that is "ready to tunnel into the dot". In Fig. 5.9 (b) the electron is inside the dot. The total charging energy of the dot is I?, = PU = e 2 / ~ and ( f the first unoccupied level of the quantum system is now e 2 / 2 c above the Fermi energy of the metal. In Fig. 5.9 (c) the electron tunneled out into the other metal and the energetic situation is restored to the initial situation in Fig. 5.9 (a). If the voltage difference between the Ferrni energy of the metal and the first unoccupied quantized states in Fig. 5.9 (a) is smaller than e 2 / 2 c the electron does not have suflicient potential energy to overcome the charging energy. In other words, tunneling into a quantum system is blocked by the Coulomb energy. At high temperatures, the thermal energy kBT broadens the quantized energy levcls. Coulomb blockade is thererore observed only for E, = e2/2c < kBT. Secondly, the quantum system has to be weakly coupled to the metals with a resistance larger than the quantum resistance h / e 2 = 26kL2. We call the energy spacing between the quantized states A E . Then the difference between two allowed tunneling energies is 2E,. + A E . In a conductance vprsus gate-voltage plot we expect a series of equidistant pcaks with high conductance when tunneling is possible and zero conductance between the peaks. Figure 5.10 (a) shows the conductance of a nanotube with tunneling contacts at low temperatures;L5,7],[5 a series of equidistant peaks is nicely seen. Their separation is given by

where a = C g / C FZ 0.1 - 0.00 1. Cg is the capacitance of the gate and C the total capacitance; a convcrts the applied voltage to the voltage drop at the nanotube. Wc can also estimate E, independently by looking at the Coulomb staircase in Fig. 5.10(b). The curve at Vg2 was measured for the gate voltage for which the zero-bias region is largest; thc onset of the current then is an estimate for E,.. In the measurement shown in Fig. 5.10 the charging energy is

95

5..? Coulomh Rlorkad~

Figure 5.10: Coulomb blockade in a metallic nanotuhe at 4.2K. (a) Conductance versus gate volt-

age showing seven equidistant peaks. Between the peaks the condi~ctmwvanishe\. (b) Current versus bias voltage at gate voltages Vgl and Vg, in (a). (c) Gray-scale plot of dl/rlVb for varying gate and bias voltage. The diamond pattern is a typical signature of Coulomh hlockade. From Ref. [S.7].

around X meV. This is more or lcss the range found in most Coulomb-blockade experiments on single-walled carbon nanotubes.["] Let us now eslimate the coherence length of electrons in metallic nanotubes from the Coulomb-blockade effect. To do so we calculate the charging energy as well as the spacing between quantized levels in carbon nanotubcs as a function of the length L.["I~[' 71 The level qpacing arises from the quantization around thc nanotube circumference plus the finite length of the tubcs. In a nanotube of length L the electronic wave functions obey the quantization condition k,, = n z / L for a givcn energy band.[5,36] We use the expression for the allowed states in carbon nanotubes as introduced in Sect. 3.3

Akl = 0 since we look at the bands going through the Fermi levcl. vp is lhe Fermi velocity for the two subbands, respectively. Then for each of graphene, which is about -6 and 7 e ~ / A band the energy spacing is X

A En = hvF-

L

[ 1.X and 2.1 meV/L(pm)].

(5.7)

The energy A E between the allowed states in the finite-length nanolube is more difficult to estimate, because we do not know the positon of the allowed k, with respect to the F e d level. Usually, simply a factor of 112 is introduced into Eq. (5.7) giving

The second quantity we need to estimate is lhe charging energy E,. for a nanotube. There are different approaches in the literature for E, .L5 'I3[' X1,[53513[5.371 The most straightforward onc is to regard thc nanolube as a finite wire of length L that is a distance b away from a conducting plate (the back gate). Then the total capacitance

96

5 Electmnic Trunspnrt

where er is the dielectric constant of the environment. With b = 100 - 1OOOnm we obtain

The relative dielectric constant contains a contribution from the tubes around the conducting nanotuhc - usually these experiments were performed on small bundles - and the SiOz layer (E, = 3.6). Since the dimension of the nanotube bundle is two orders of magnitude smaller than the thickness of the Si02 layer our best cstimatc for E, is the oxide value. Then E,. is roughly 2.5 meV pet nanotube p m length.[' 51 We insert our estimates for charging energy and level spacing in finite tubes, Eqs. (5.10) and (5.8), into Eq. (5.5) and obtain the separation of the conductmcc pcaks

The last relation follows from Vg = 50mV which can be read from the peak spacing in Fig. 5.10. We thus obtain from thc Coulomb-blockade effect a coherence length for the electrons between 100nm and several ym. In the experiment performed by Nygkd et UZ.,['.~~ which is shown in Fig. 5.10, the spacing between the bias electrodes was 500nm, which is in reasonable agreement with the length we estimated from the Coulomb charging. It should be kept in mind that Eq. (5.1 1) is really a very rough estimate. In the literature values of 2'] L = 3 - IOpm are often deduced from the Coulomb-blockade effect.['

5.4 Luttinger Liquid Up to now we discussed electronic transport in metallic nanotubes in a semi-classical picture. The only difference from three-dimensional systems we explicitly used was the peculiar band structure of carbon nanotubes, which were then decribed as ordinary metals. By this we implicitly treated the interacting electrons as quasi-particles. Theoretically, the effect of electron-electron interaction in one-dimensional metals has been studied for decades. It was found that the quasi-electron description failed for an arbitrarily weak intcraction between the electram. The resulting state is commonly called a Luttinger l i q ~ i d . [ " ~ ~ 1 , [ " ~ ~ 1 Luttinger liquids have a number of exotic and fascinating properties. The known example is spin-charge separation. The original model proposed by ~ o m o n a ~ aand l ~I+uttinger15 .~~ 3'1 was a one-dimensional linear electronic dispersion for the non-interacting electrons. Once a finite interaction is switched on between the particles, an electron is no longer a stable excitation. It decays into spin and charge plasmons. It turns out that these two types of plasmons havc different velocities, i.e., with time the spin part and the charge part of the electron separate in space. As we know, armchair nanotubes have a linear dispersion around the Fcrmi lcvcl with only two Fermi wave vectors kF = 2 r / 3 a and -kF. Moreover, the electronic transport in the tubes is ballistic over long distances as we saw previously. Armchair nanotubes are thus an ideal system to study a theoretical idea first proposed in 1950.~~,~'1 Naturally, Luttinger-liquid hehavior in carbon nanotubes was studied theoretically soon after their disco~ery.["~~1~["~~1 We will not descrihe the model here; instead we focus on some

5.4

Luttinger Liquid

97

Figure 5.11: Conductance of a metal nanotube in tunneling contact with two metal electrodcs a\ a function of temperature. Full lincs arc the measured curves on lwo different nanotubes. The brokcn lincs werc obtained after considering the temperature dependence of the Coulomb-blockade effect. A power law is seen as a straight line on a log-log plot. A schematic picture of the experimental setup is shown in thc insct. From Ref. [S.43].

prcdictions made for Luttinger-liquid behavior in carbon nanotubes and experiments confirming these predictions. A very nice review o r this topic in both theory and experiment can be round in Ref. [5.42]. An experimental verification or the Luttinger state, which is undoubted from the theoretical point of view, involves two basic questions. Can a nanotube be shown to behave like a Luttinger liquid? And can the most exciting properties like spin-charge separation be proven experimentally? While the first question was addressed for nanotubes in a numthe second one is still open. Nevertheless, existing experiments ber of st~dies,[~~~~~["~~]~-["~~] give some good evidence that a Luttinger liquid is realized by single-walled carbon nanotubes. To confinn Luttinger-liquid behavior most experimental studies measured the conductance as a function of temperature. The conductance is expected to show a power-law dependence on T , G T", when tunneling from electrodes into the nanotubes limits the transport properties. We discuss the power-law exponent a in detail below. Figure 5.1 1 shows the conductance of a junction bctween metals and a metallic nanutubes as a function of temperature on a doublc logarithmic scale.[5471 Full lines are the raw data. Broken lines were corrected for the tempcraiure dependence of the Coulomb-blockade crfect, which sets in at low temperatures in the experiments. The dashed lines follow quite nicely a power-law dependence ovcr the entire temperature range. An exponent a 0.3 was measured on diffcrcnt nanotubes by Bockrath et al.rs.431 They also showed that the dependence of the conductance on bias voltage followed a universal scaling law, which was predicted for 15.231, [5.401, [5.421, [5.431 Luttinger-liquid behavior in carbon nanot~~bes. The observation of the powcr law in the tunneling measurements is in good agreement with the theory of a Luttinger liquid. However, it was pointed out recently by Egger and ~ o g o l i n [ " ~that ~ ] the data of Rockrath et a/.[5.431may not provide an unambiguous proof for the Luttinger state. As already said, the Coulomb blockade itself exhibits a temperature dependence that in the above experiment was accounted for using a standard r n ~ d e l . ~ ~Egger .~"] and Gogolin instead proposed that the power-law dependence in Ref. r5.431 could be due to nonconventional Coulomb blockade.[w4"] In the above description of the G -- T" we did not specify the exponent a that depends on the Luttinger parameters g and, interestingly, on the geometry of the contact. If an electron tunnels into the open end of a nanotube (the caps are

5 Elrctronic Transport Figure 5.12: Plot of &,d versus abt,lk within thc Luttinger liquid (LL) theory; both exponents depend on g as givcn in Eqs. (5.12) and (5.13) (full line). The broken line corresponds to a fixed ratio of two between the two a>.Data points are measurements of the end- and side-tunneling exponent hy BockNygird p t a1.,15,71 Yao et ul.,15441 rath et al.,[5.43J and Postma et 451

neglected) the exponent &nd is larger than q,"lk for tunneling into the side of a tubes. That &,,d is larger than ab,lk means that the tunneling density is more strongly suppressed in the end-tunneling case. This is not specific to Lutlinger liquids, and we can understand it with a hand-waving a r g ~ r n e n t . ~In~the .~~ end-tunneling ~ configuration, electrons can only move in one direction to accomodate the additional electron. In contrast, in the side-tunneling experiment they can spread to the left and right. Thus, e-g., for Coulomb blockade a similar difference is found with & n d / ~ l k= 2.15.461The important difference for a Luttinger liquid is that this ratio is always larger than two. The Luttinger parameter g determines the exponents

with

E, and A E are the charging energy and the level spacing introduced in Sect. 5.3. Using the estimated values givcn there we obtain g x: 0.4; typically the expected range is g = 0 . 2 0.4.["2"17[5.401Note that the small values of g are the limit of strong electron correlation; g = 1 corresponds to the non-interacting F e m i gas. From Eqs. (5.12) and (5.13) follows

which is always larger than two because g ic; pmitive and smaller than 1. A possible verification of the Luttinger liquid in carbon nanotubes is thus to demonstrate that the exponent for end tunneling is more than twice the exponent for side tunneling. The first attempts to measure the different tunneling rates and their temperature dependence used nanotubes on top of the electrode as side-tunneling systems.15.71,15.431 For an endtunneling measurement the tubes were deposited on the SiOz layer hefore depositing the electrodes. This process was found to cut nanotubes into two unconnected segments. Although

5.5 Summary

v

C3 A

r

Segment I Segment II Across the k~nk

Figure 5.13: Experimental measurement of the exponents for end and side contacts. (a) AFM picture of the single-walled carbon nanotube used in the conductancc mcaxurements. The tube lies across four metal electrodes and has a sharp kink between the sccond and third electrodes. Such a kink can result from a pcntagon-heptagon defect as in (b). (c) Dependence of the conductance on temperature for side tunneling (scgmcnt I and 11) and a measurement acroxs ihe kink. From Ref. [5.441.

both samples showed a power-law dependence on temperature the ratio between thc two exponents was very close to two. In Fig. 5.12 we plotted &,d versus a , l k by the Cull line. The dashed line is for a ratio of 2; the two experiments clearly fall onto this line. The measure~ ]Postma et in contrast, both agree very well with the ments by Yao et a ~ . [ " ~and predictions from Luttinger-liquid theory. In both experiments an intra- or intertube junction waq used for the cnd tunneling. Figure 5.13 (a) shows an AFM picture of the nanotube used in conductance measurements by Yao et al.LS 441 The tube lying on top of four metal electrodes has a sharp kink between the second and third electrode. In Fig. 5.13 (b) a possible atomic structure of a nanotube with a kink is displayed. This kink or break can be viewcd as a weak contact between two Luttinger liquids (thc nanotube on both sides of the defect). The power-law exponent is then simply , otherwise the same results are expected.154XI In the experidoubled q , d d,, = 2 ~ , d but mcnts end tunneling is thus measured across the kink, whereas side tunneling is measured in the usual way, p.g., between the uppermost and second electrode from the top. The cxperimental data are depicted in Fig. 5.13 (c); clearly end-end tunneling has a different dcpcndence on temperature than side tunneling (segment 1and TT). Plotting the obtained exponents in Fig. 5.12 reveals that indecd the ratio between the two cxponents is larger than two. There is thus quite some evidencc that a Luttinger liquid was observed experimentally in single-walled carbon nanotubes. It is a challenge for the future to explore the different effects predicted for this strongly correlated one-dimensional state.

5.5 Summary Tn this chapter we looked at the transport properties of single-walled carbon nanotuhes. We saw that armchair tubes are ballistic conductors over 100nm to pm length. With highly trans-

100

5 Electronic Transport

parent contacts a conductance close to the hallistic limit was observed experimentally. This was explained theoretically by two arguments. First, the potential of a possible scattering center has to be averaged over the nanotube's cirurnference. Secondly, symmetry forbids a mixing of the two conducting states, which makes armchair nanotubes quite robust to external pertubations. In semiconducting tubes, on the other hand, the mean free path for elastic scattering seems to be shorter. Quasi-ballistic transport was reported with a mean free path around 100nm. The uppcr limit of the current a nanotube can carry is set by electron-phonon coupling. Accelerating electrons to a high energy causes cfficient relaxation by emission of optical or zone-boundary phonons. The resistance o l a nanotube therefore depends linearly on the applied voltage along the tubc. At low temperatures Coulomb blockade was observed in single-walled carbon nanotubes. Tunneling into the nanotubes is then suppressed by the Coulomb interaction until an additional potential energy is provided to overcome the charging energy. Finally, we discussed a couple of cxperiments that indicate that armchair nanotubes behave like Luttinger liquids.

6 Elastic Properties

Nanotubes are small tubes. As such, we can attempt to describe their mechanical properties in a continuum approach. In other words, one may ask whether the nanotubes behave like macroscopic pipes as regards, e.g., their cross section when stretching them. More generally, we investigate the well-known stress-strain relationships for nanotubes from classical mechanics; it turns out that such a procedure works quite well. We emphasize that it is by no means obvious that such an approach should work; it has to be tested against the actual behavior of the tubes as determincd from experiments or from microscopic calculations taking the atomlc structure into account explicitly. An example where simple intuition about the nanotube having a homogeneous wall fails was rcccntly given by Dobardiii et ul.,['~'] who showed that the radial breathing mode in nanotubes contains a small r-component; only armchair tubes (8=30°) are strictly radial. Their argument, which was based on symmetry, was confirmcd also in oh-initio calculation^.[^.^] A recent extensive review of the mechanical propcrties of carbon nanotubes was published by Qian ct al. ["I Experiments that tell us about the stress-strain relationship, i.e., the various moduli of the tubes, arc, e.g., bending,Lh41.[" 51 indentation16.61or resonantly vibrating beam experimcnts,[6.7] micro-Raman measurements of nanotubes cmbedded in epoxy,[' 1' and measurements of the vibrational frequencies under hydrostatic pressure.[""]-IU2l Hydrostatic pressure changes the volume and in anisotropic systems - such as nanotubes - also the shape of the unit cell. Studying the structural or vibrational propertics thus provides valuable insight into the elastic propcrties of the material. As we shall show in this chapter, the continuum approach works well, even for some very small tubes investigated. We shall first work through the continuum model of the tubes and derive a number of expressions based purely on gcomelric quantities, such as the inner or outer radius of the tube (Sect. 6.1). We compare the analytical, classically obtained results to ah-initio calculations of the same quantities. Here, and in the experiment, we find good agreement, making the continuum model a straightforward and reliable way to describe the elastic properties of carbon nanotubes. Finally, we present some results on micro-mechanical manipulations of nanotubes (Sect. 6.3).

6.1 Continuum Model of Isolated Nanotubes The mechanical properties of solids have been well studied and are described in textbooks on classical mechanics, most comprehensively probably by Landau and ~ifshitz.["l"] The general relationship between a macroscopic stress CJ applied to a material and the resulting

102

6 Elastic Properties

microscopic deformation E is usually described with the elastic-constant tensor ~

l

~

or, in index notation (i, j, k, 1 = x , y,z, and repeated indices arc to be summed over),

The symmetry of a crystal restricts the number of - in general 8 1 - non-zero and linearly independent constants Cljkl.For example, in nanotubes obviously C,, = Cym,y,since the tubes are isotropic in the plane perpendicular to the z-axis. The elastic constants can be measured by a variety of techniques like ultraqound experiments or dircct measurements of the lattice constants under stress. Despite the large interest in the mechanical properties of carbon nanotubes, in particular as reinforcement materials,["5] successful measurements of the elasticity constants have not yet been reported. Young's modulus E was investigated several times both thcoretically and experimentally; it was generally found to be on the order of thc value of graphite.[6.71,[6.161-r6.221 For some selected tubes elastic constants were calculated with an empirical force-constants model by ~u.l"q,[' 241 RU['.~'] included in their model for multiwall nanotubes the van-der-Waals interaction between neighboring laycrs. In the continuum model presented here we find the elastic response of nanotubes under pressure differently. The nanotube - in this model - is looked at as a hollow cylinder with closed ends and a finite wall thickness made of graphene. This approximation, in addition to yielding the deformation of a nanotubc under pressure, provides an insight into the question whether the differences in elastic properties of graphite and carbon nanotubes follows plainly from their different topologies or from additional physical effects, e.g., rehybridization. We expect such a rehybridization to become more noticeable for smaller tubes. The starting point of the continuum-mechanical description of the elastic properties of a tube with finite wall thickness is thc equilibrium condition

The generalized Hooke's law in an isotropic where q ;ire the normal co~rdinates.["""I.[~.~~~ medium is given by

or - the inverse relationship -

where E is again Young's modulus and v Poisson's ratio. Young's modulus is defined as the ratio of applied pressure and strain, E = p/&, and Poisson's ratio of lateral contraction to The strain &ik is defined by the displacement vector u longitunal stretching, v = -&&/,., describing the shift of a point in the deformed material

.

6.1 Continuum Morlc~lof Isolated Nanotuhes

Figure 6.1: Continuum-mechanical model of a nanotube - a closed cylinder oT length I with inner rdius K i and outcr radius I?,. Thc boundary conditions under hydrostatic pressure are indicated

in thc figure; A is the ratio between the cap arca and the area supported by the cylinder walls, A = R;/(R: - R;). inserting Hooke's law (6.4) into Eq. (6.3) the fundamental equation of continuum mechanics can be derivedr6.l3]

Since the rotation of u vanishes in our problem, Eq. (6.6) in cylindrical coordinates simplifies to I d(ru,) 1 au8 au, divu = -- =const.

+--+ r

r

&

Figure 6.1 defines the parameters in the continuum appmximation of a single or nlultiwall nanotubc. The tube is modcled as a finite hollow graphene cylinder of length 1 with closed capsand inner and outer radius Ri and R,, respectively. At z = O, in the middle of the tubc, the displacement u, = 0 and increases continuously in and -z directions, E,, = au,/dz = const. The circumferential displacement ufi is independent of 6 , since our problem is rotationally symmetric, i.e., d u 8 / d 6 = 0.Finally, according to Eq. (6.7) 1/ r . d ( r u , ) / d r is again constant. Thc strains E,, in cylindrical coordinates are therefore given by1"271

+

a&

-

h r2

1 dua r da

U,

h

r

r2

du,

E,,=-=const. (6.8) dz The constants a, I?, and E,, are determincd by the boundary conditions for s under hydrostatic pressure. Assuming that the pressure medium cannot enter the nanotube cr,,(R,) = O and o,,(R,)= - p , where p is the applied (hydrostatic) pressure. Along the z-direction the pressure transmits a force -p . ZR: on the caps of the tube (cap area=nR;). The area supported by the wall of the tube is K ( R ; - R:) and hence o, = - R ~ / ( R :- R;) . p = - A p . Inserting the boundary conditions into Eq. (6.4a) we obtain the constants of integration in Eq. (6.8) Err

= -- a - - ,

ar

~~~,g=--+-=a+-,

and

Reinserting them into Eq. (6.8) finally yields the strain tensor in a cylindrical tubc under external hydrostatic pressure p

104

6 Elastic Properti~s

Thc mixed strain conlponents &,j, (i $r J), vanish. The strain tensor in a nanotube under prcssure, in this model, is thus given by two clastic constanls E and V , and the geometry of the cylindcr. There are some general implications for the strain tensor in the continuum approximation. The change in the lube's length or thc translational periodicity along thc z-axis is described by E,,, while ~,9,9 is the radial or circumferential deformation. Both components are always ncgative for posilivc pressure. Moreover, the circumferential deformation is always larger than the axial deformation

as expected for an anisotropic system. Note that the differences in &?,9, and &,, are only a consequence of the cylindrical geometry as thc nanotube wall is taken to be isotropic. It is also straightforward to adapt the model to different physical situations. Multiwall tubes may be described by choosing the inncr and outer radius according to the number of layers of a given tube. Open nanotubes may be described by modifying the boundary conditions used to find thc integration contstants approprialely. One can even address the question as to whether or not thc pressure medium enters the nanotubes.lb I The change in the wall thickness, E,,, can be posilive or negative. In particular, it depends on r, and there is a radius ro wherc Err(r0)> 0. For isolated single-walled nanotubes a varying wall thickness is not very meaningful, since it consists of only one graphene sheet. We choose ro = v ) / ( l - 2 v ) Ri, i.e., the radius where &,,(r0)= 0. However, in multiwall tubes a natural choice is the mean value of Ki and R,, and for reasonable values of r, the strain do not depend very much on r in single-walled nanotubes. components err and The strain components that are responsible for the experimentally observed frequency shift of the high-energy Raman modes are the circumferential and tangential slrain components. As discussed in Chap. 2 the high-energy vibrations are parallel to the nanotube's wall. Hence E,, is negligible both Tor single and mulliwall nanotubes. Whal follows for the strain in a nanotube from our continuum model? Consider unwrapping the tube to a rectangle; the strain along (he narrower, circumferential direction is then ~ $ and 8 the longer side is deformed according to E,,. With typical values for the radii and the elastic constants of single-walled nanotubes (Ri = 5.2& Ro = 8.6& E = l ~ ~ a - and l, v = 0 . 1 4 ) 1 " ' " ] ~ [ ~.Etgt9(p) ~ ~ ] = - 2 . 0 4 ~ ~ a - I pand &::(p) = - 1 . 0 7 ~ ~ a - 'Within ~. the elasticcontinuum model the ratio between circumferential and axial strain is thus 1.9, the unit cells of the tubes under hydrostatic pressure deform as shown in Fig. 6.4. As a consequence, the sixfold hexagonal symmetry is lowered, and, r.g., the E2g vibrational modes degenerate in plain graphite should split under pressurc. Another consequence is that the radial modes should have at most twice the pressure coefficient of that of the axial modes. As we show later, this is not observed experimentally; the radial modes in bundled tubes are strongly affected by the bundling, a property that we have so Fir not included in thc model. We are now in a position to calculate from our strain pattern the change in phonon frequency under pressure from a linear expansion of the dynamical equation in the presence of strain. Before doing so, we compare the results of the elastic-continuum model to other, more sophisticatcd calculations of the elastic properties of carbon nanotubes.

''

d(l+

6.1

105

Continuum Model o f Isolated Nanotuhes

Figure 6.2: Ab-initio calculat~on of the axial (closcd symbols) and circumferential (open symbols) \train in single-walled nanotube bundles. Circlcs refer lo an (8,4), up trianglcs to a (($3, and

down triangles to a (10,O) nanotube. The full lines show a least squares fit of the $train components in the three tubes; the brokcn lines are the continuum approximation [sce Eq. (6.1 O), r = 4.05 A].

,

b

0

1

2

3

4

5

6

7

8

9

Pressure (GPa)

6.1.1 Ab-initio,Tight-binding, and Force-constants Calculations There are general approaches for finding theoretically the elastic properties of a material. Either thc lattice constants under stress are calculated by directly incorporating the applied stress tensor into the calculation or the elastic constants are found from the sccond derivatives of the energy in a strained unit cell. The fonncr approach was used by Reich et nl.Lb.28]in ub inirio calculations as well as by Venkateswaran ~t a1.[6.91in tight-binding molecular dynarnics; whereas ~ u [ ~ . ~ ~ 1 7and [ " ~Robertson J rt used the latter in their force-constants calculation. Ah-initio calculations of three small-diameter nanotubes, an armchair (6,6), a zigzag (10,0), and a chiral (X,4) nanotubc (chiral angles 30°, O", and 15"), are shown in Fig. 6.2. The circumferential &fie= [r(p) - r,]/r, and axial strains E,, = [ a ( p )- aa]/aa are given by thc stress-dependent radius r(p) and the translational periodicity a ( p ) ;r, and a, are the ambient pressure values. As in the continuum approximation the circumferential strain is larger than the strain along the nanotube axis. Moreover, the strain components are found, to a very good approximation, to be independent of the chirality of the nanotubes, a parameter completely neglected within the continuum approximation but justified a postrriori. The calculated pressure slopes of the radial and axial strains are E = - 1.5 ~ ~ a - and l p E, = -0.9 ~ ~ a - (full ' p lines in Fig. 6.2). This is in excellent agreement with the elastic-continuum model ford = 0.8 nm aq can be scen in Fig. 6.2, where the strains obtained from Eq. (6.10) &fie= - 1.42TPa- l p and F, = - O . X ~ T P ~ - ' are ~ shown as broken lines. Thc similarity between the elastic-continuum model and the first-principles calculations for such small tubes is quitc remarkable. Thc continuum-mechanical approximation works well even in the limit of a single atomic layer and a strongly curvcd surface. Larger tuhcs, such as the typically studied 10-A-diameter tubes should also be well described by the continuum model. ~ u ~calculated ~ . ~the~entire ] sct of elastic constants CiJklfrom force constants fitted to reproduce thc phonon dispersion in nanotubes. From these elastic constants for a two-layer ( 1 0,lO) tube the strain components may be obtained from Eq. (6.2); they arc given in Table 6.1. Thc strains in both directions are somewhat smaller than within the continuum model. Thc ratio hctween the axial and circumferential strain is predicted to be 3.5. Vcnkateswaran et 01. [' y] performed molecular-dynamics simulations of (0,9) single tubes and bundles of tubcs under

106

6 Elastic P m p ~ r t i ~ s

Figure 6.3: Molecular dynamics simulation of the axial (closed circlcs) and the circumrerential (opcn circles) strain under hydrostatic pressure. The data points are taken from Fig. 4 of Ref. 16.91. The circumferential deformation shown here corresponds to the average of the two radii under pressure reported by Venkateswaran et a1.[6,91

:E?,,

= - 3.41 ~ ~ a - ' p

%

-

0

1

2

3

5

4

Pressure (GPa)

pressure. The norrnalizcd axial and circumferential strain they reported in the bundle is shown in Fig. 6.3. The compressibility along the axis is similar to the continuum value, whereas the circumrcrential strain was found to be much larger than in the continuum approximation. Despite some differences in the absolute values of the predicted strains all calculations agree in the following fundamental points: under hydrostatic pressure the circumferential strain is larger than the axial strain by a Factor of 2-4, and the linear compressibility along the nanotube axis is similar to that of graphite ( - 0 . 8 ~ ~ a - ' p ) . An experimental determination of the elastic constants on isolated nanotubes is almost impossible. Methods like X-ray scattering under external stress cannot be applied to isolated tubes, because nanotubcs strongly differ in their translational periodicity. There are many studics of Young's modulus and nanotubes under tensile load, but the few for single-walled nanotubes were performed on bundles, see Sect. 6.3. The change in hond lengths and angles can be deduced from Raman scattering. The high-energy in-plane vibrations of single-walled tubes are most sensitive to the a bonds, whereas they arc only weakly affected by varying the intertube distance. We include here results on the high-energy modes obtained on nanotube bundles, because of these modes' sensitivity to the G bonds.

Table 6.1: Axial and circumferential strain under hydrostatic pressurc in the four approximations discussed in this chapter. Thc first three rows are for radii typical for single-walled nanotubes. The next two rows demonstrate the exccllcnt agreement between the clastic-continuum model and ab-initio calculations. The experimental value for graphite is listed Tor comparison.

Continuum model Elastic constant^[^.^^^ Molecular dynamicsl"'l Continuum model ~b inirin[6.301 ~ra~hite[""]

r(A) 6.9

6.8 6.1 4.05 4.05

d ~ d 1/9d p ( ~ p a - )l -2.04 -1.74 -3.41

d&zz/dp(TpaZ1 ) - 1.07

-0.49

&1919/%

1.9

-0.91

3.5 3.7

-

1.42 -1.5

-0.86 -0.9

1.6 1.7

-0.8

-0.8

1 07

6.2 Prrssure Dependence ofthe Phonon Frequrnries

6.2 Pressure Dependence of the Phonon Frequencies Strain changes the phonon frequencies because the bond length and angles are different in a strained crystal. In the following we first assume that the strain tensor of thc tube is known, to demonstrate how to find the phonon frequency shift for an arbitrarily strained tube. We then examine the case of a hydrostatic strcss tensor, since many Raman experiments were performed under hydrostatic pre~~~re.16~51~[6.12]~["7J,[6~321 From a known strain tensor the phonon frequency shifts follow from the dynamical equation modified to include terms linear in strain via the phonon-deformation potentials.[6.33]-l"3hl The phonon-deformation potential relating the volume change of the unit cell with the frequency shift under pressure is called the Griineisen parameter. We show in the following how to derive the vibrational frequencies in nanotubes under pressure by malung reference to the rolled-up graphene sheet. The basic idea of the approach is depicted in Fig. 6.4. The figure shows schematically a (6,6) w d an (8,4) tube under exaggerated hydrostatic pressure (corresponding to E 100GPa). Because of the larger radial than axial strains, hydrostatic pressure changes not only the area of the graphenc hexagons but also distorts their shape. The sixfold hexagonal symmetry is broken under pressure, which splits the doubly degenerate E2g graphene optical modes into a higher and lower frequency component vibrating parallel and perpendicular to the higher strain direction, respectively. For nanotubes this corresponds to a stronger pressure dependence for a phonon eigenvector with the atomic displacement along the circumferential direction than for an axial vibration. A similar analysis can be performed for other strcss tensors like, e.g., a uniaxial load.lh.151,[6.37] We perform a quantitative analysis by unwrapping the tube to a narrow graphenc rectangle. Thc strain in the graphene sheet due to &flly and E,, reads (after transformation to the principle axis of graphene)16.12]

separated into the hydrostatic and nonhydrostatic components

where 0 is the chiral angle. The deformation of the graphene unit cell is only hydrostatic if '77

=€

1 1%. ~

Figure 6.4: Schcmatic of the distortion of a (6,6) and an (8,4) nanotube under hydrostatic pressure, i.e., = 2 ~ , , The . strain is fully symmetric in the point groups of the tubes, but not for the graphene hexagon. The hexagon area as well as its shape are altcred hy applying pressure to a nanotube.

(696)

"(

Table 6.2: Phonon-deformationpotentials for nano- 'I2)

1.24 0.4 1 ~ ~ ~ ~ h i t ~ 1 36" . 3 ' 1.59 l % [ ~ 0.66 Diamond ( c ~ b i c ) [ ~ SO ~ ~ " ] l 0.5 1

tuber (tight-binding calculation), graphite (erpcriment and ah-inirio calcula~ion),and diamond (experiment). The shear-defnrmatron potential in diamond is for an applied uniaxial qtrcss along the (001) direction.

We expand the dynamical equation to tcrms linear in strain to find the phonon frequencies in the strained graphene cell,[h~"l,~6.361

whcre v is the atomic displacement, rn the reduced mass of the atoms, and the strain-free frequency. The second summand describes the change in phonon frequency due to the strain; Kikml= aKik/d&kl is the change in the spring constanls of the strained crystal. The syminetric tensor K(') has only threc non-zero components because of the hexagonal symmetry of the graphene sheet, r ~ a r n e l ~ , [ ~ . l ~ ]

From thc dynamical equation (6.14) we obtain a secular cquation with the help of the tensor components in Eq. (6.15) and the strain in the graphenc sheet (6.12)[6.331.[6.3h1

where A = m2 - W; is the difference between the squared strain-dependent frequency o and the Irequency in the absence of strain @. Diagonalizing Eq. (6.16) thus yields the relative shift or the phonon energy in the strained graphene ~ h e e t [ ~ . ' ~ l

In Eq. (6.17) two phonon-deformation potentials relate the frequency shift with strain. The first deformation potential ( K I 1+ R 1 2 ) / 4 w ;= -y is the Griineisen parameter, which describes the frequency shift for a hydrostatic deformation of the graphene hexagon. The splitting of the modes undcr shear strain arises from the second term. Measured and calculated deformation potentials in graphite, carbon nanotubes, and diamond are given in Table 6.2.

6.2 Pres~ureDqwndence of thr Phonon Frequenricls

I . . . " l . . . . I l . . . l 1500

1600

1700

Raman Shift (em-')

1800

0

2

4

8

lo

Pressure (GPa)

Figure 6.5: Raman scattering on carbon nanotubes under hydrostatic pressure. (a) High-energy range of the Raman spectrum and pre5sures up to 9 GPa. (b) Pressurc dependence of thc phonon frequcnc~es as measured in (a). The numbers give the logarithmic pressure slopcs in TP~-' . Thc error of the slopes is 0.1 T P ~ - ]unless statcd otherwise. From Re[. 16.431.

Interestingly, the frcquency shift is independent o r the chirality of the nanotuhc, i.e., the way the graphene rectangle is cut and strained. A mode vibrating parallel to &fie,the highstrain direction, always shifts according to the plus-solution of Eq. (6.17), whereas vibrations parallel to E,, have a frequency shirt below thc hydrostatic cwtribution. Between these two limiting cases the phonon modcs show a dispcrbion sin~ilarto, e.g., the dispcrsion in wurzite crystals. The dependence of the phonon frequency on thc displacement direction of the eigenvector in strained materials is discusscd in two papcrs by ~ n a s t a s s a k i s . [ ~ ~ ~ ] ~ [ " ~ ~ ~ Raman scattering on strained tubes was mostly performed under hydrostatic pressure. To calculate E,, and E O for ~ this situation we use the circumfercnlial and axial strain components obtained from elasticity theory and the elastic constant calculations by ~u.["~4] With lhe phonon-deformation potentiitls for graphene and nanotuhes as given in Table 6.2 we find the hydrostalic component of the frequency shift

and a shcar strain splitting

Figure 6.5 (a) shows the typical high-energy Raman spectra of single-walled nanotubes 1 , a[ ~review . ~ ~ ]of , [ thc ~ ~ experimental ~~ work on under pressures up to 9 ~ ~ a ; L " ' ~ ~ 1 ~ . ~ ~ for pressure, see Ref. [6.44], for a discussion of the Raman peaks under ambient conditions see Chap. 8. Under increasing pressure the phonons shirt to higher frcquency. At the same time the Raman peaks lose intensity and broaden. The origin of the intensity loss is not fully

h

Elastic Pn~perties

Figure 6.6: (a) A high-energy cigenvector of an (X,4) nanotube with a Crequcncy of I505 cm-' . The atomic displacement is parallcl to the circumfcrence. (h) A1 high-energy eigenvector of a (9,3) tube (1627 cm-' ). The displacement is parallel

to the carbnn-carbon bonds. The direction of tbc helix in both tubes, which is obtained from thc screw-axis operation, is indicated by [he gray lines.

undcrstood; it is attributed to an hexagonal distortion or a squashing of the tubes at high 461 A similar washing out of thc nanotube pcaks is also found in highpressure optical a b ~ o r ~ t i o n . [The ~ . ~high-energy ~I Raman frequencies as a function of pressure are shown in Fig. 6.5 (b). The numbers arc the normalized pressure slopcs in Tpa-I. All four of them are in excellent agreement with the expected hydrostatic component of the frequency shift. The predicted shear-strain splitting of 0 . 6 ~ ~ a -in l , contrast, is completely absent. It was observed in high-pressure Raman experiments with an excitation wavelength of 647 nm. Thc different behavior at different excitation energy can be attributed to metallic nanotubes, which are resonantly excited in the red energy range (Sect. ~ . 5 . 2 ) . ~ ~For . ' ~ 1the 50-called semiconducting spcctra as in Fig. 6.5 (a), however, only the hydrostatic stress component is found experimentally. The possible explanation for the absence of the shear-strain splitting lies in the phonon eigenvectors of the high-encrgy vibrations, see also Sect. 2.3.5. In achiral armchair and zigzag tubes the eigenvcctors are completely determined by the high symmetry of the tube. In particular, the A, phonons in achiral tubes are either polarized along the axis or along the circumference. Since experimentally only scattering by A 1 phonons yields a noticable Raman intensity thc high-pressure spectra should show the shear-strain splitting if the sample consists d achiral tubes. In contrast to the higher-symmetry achiral tubes the phonon eigenvectors of chiral tubes arc not fixed by symmetry because of the missing mirror plancs. A distribution of displacement directions with respect to the circumference or the tube axis washes out the Only thc hydrostatic splitting introduced by the shear deformation in Eq. (6.17).L6 componcnt is then observed experimentally. 919[6451-[6

1213r6.411

Figure 6.6 shows the Al high-energy modes of (a) an (8,4) and (b) a (9,3) nanotube as ~ ~ ~black ~ ~ ticks indicatc the displacemcnt obtained from a first-principlcs c a l ~ u l a t i o n . [The dircction of the atoms. A high-energy A1 mode always has an eigenvector where the two carbon atoms in the graphene unit cell move in opposite directions; moreover, thc displacement is constant when going around the circumferencc of the tube. The atoms in Fig. 6.6 vibrate along the direction of the carbon-carbon bonds and not along the axis or the circumference. This is best seen for the ( 9 3 nanotubes in (b). The calculated dependcnce of the eigenvectors on the direction of the carbon bonds thus very nicely explains the high-pressure Ramm experiments. The good agreement between the theoretical and experimental hydrostatic pressurc

Figure 6.7: (a) Doubly degenerate El eigenvector of an (8,4) tube with a frequency of 161 0 cm-' . The sequence shows the change in displacement when going around the tube in steps of 32". The atoms highlighted by the small circles are connected by the screw symmctry of the tube. (b) ?-component of the displacement versus the xy-componenl. The strongly varying displacement direction as shown in (a) results in an open displacement ell~pse.After Ref. [6.48].

compnncnt, on the other hand, confirms the validity of the continuum model for single-walled carbon nanotubes. Let us take a brief look at the Em phonons in chiral nanotubes. For these modes the atomic displacement shows 2m modes when going around the circumference. Although E l and Ez vibrations are Raman active they do not contribute strongly to Raman spectra as seen in Fig. 6.5 (a). This was found experimentally on isolated and unorientcd tubes (Sect. 8.4) and is understood by the depolarization effect in carbon nanotubes as explained in Chap. 4. Ncvcrtheless, the E modes show an interesting dependence on the azimuthal angle 29 that we want to include here. In Fig. 6.7 (a) we show an E l eigenvector of an (&4) nanotube. The nanotube is successively rotated by 32" to show how the eigenvector evolves when going around the nanotuhe. As expectcd, the magnitude of the atomic displacement (the length of the ticks) is modulated by a sin cp function. Contrary to what is generally expected, howcver, the direction of the displacement varics as well. Whereas the displacement ol' the highlighted atom is perpendicular to one of the carbon-carbon bonds in the first picture they are almost parallel to the bonds in the last two pictures. The angular dcpcndence of the direction of the cigcnvectors becomes more obvious when plotting the atomic displacement along the z-axis versus the circumferential displacement [Fig. 6.7 (b)]. Open ellipses in such a plol describe E symmetry eigenvectors with a varying or "wobbling" displacement direction. The open ellipse that makes an angle of = 30" in Fig. 6.7 (b) corresponds to the eigcnveclors shown in (a). The degenerate eigenmode has the same ellipse. Two el1ipr;esperpendicular to each other represent modes of the same symmetry but different freyuencics. The diagram in Fig. 6.7 (b) shows both the 1610 and the 1591 cm-l El cigcnvector.

6.3 Micro-mechanical Manipulations Most of the experimental work on elastic properties has been performed on multiwall nanotubes, which are no1 in the scope of this hook. Nevertheless, we show also some nice pieces

Figure 6.8: Sword-in-sheath experiment showing (a) a multiwall tube loosely connected to two AFM tips. (b) Samc tube after stretching and pulling out the inner tube (now curled up) and leaving thc outer one bchind (or vice ver,su). (c) Schematic of the tclcscope idea. After Ref. 16.491.

Figure 6.9: (left) (a) Schematic of thc realization of a micro-electromechanical device. A multiwall nanotube bearing anchored by Al and A2 is devised to hold a mirror that can bc rotated via a stator clcctrode S l , S2, and S3 (buried underncath the surface). (b) Scanning electron lnicroscope (SEM) imagc of the realization of the device, just bcforc etching the mirror. (right) Sequencc of SEM images of [he mlrror plate at different angles; the angular pwition of the mirror is indicated by the bars undcr the images. The nanot~thcruns from top to bottom in the images. After Ref. r6.511.

of work that involve nanotubes sliding within each other in a multiwall arrangement, a socalled sword-in-sheath experiment. In Fig. 6.8 from Yu et al.16.491we show how a multiwall nanotube atlachcd to the tips of two opposing atomic-force microscopes is pulled apart. In Fig. 6.8 (a) thc two microscope tips have not yet stretched the tube, it still has somc slack. In (b), after tearing the tube, we see the two remainders of the broken nanotube. The part on the upper tip has curled up, the lower rest of the nanotube still stands up straight. During the tearing process the measured Corccs give information about the shear strength between the two parts of the tube. We would expect them to he on the order of the in-planc shear strength of graphite, i.e., relatively weak. Yu et ul. found a shcar strength between 0.1 and 0.3 MPa, which is indeed similar io graphite. Cummings and ~ettl['"] in a similar experiment pulled and pushed the inner part oC a multiwall nanotube in and out repeatedly and found no wear over ahout 20 cycles. Later, Fennimore et ul.l'.sll extended the work on bearings to a fascinating electromechanical device that had a multiwall nanotube bearing for a small plate that can be thought of as acting as a tiny mirror, see Fig. 6.9. This device was integrated on a Si chip and worked over repeatcd rotations, The authors expect their device to work to quite high frequencies due to the small involved mass and a relatively high torsional forcc constant.

Figure 6.10: (a) Atomic-force microscope image of a bundle of carbon nanotubes supported over a mask in a clamped-beam configuration. From the experienced forces thc clastic modulus oC Lhe nanolube rope can be derived. (b) Schematic (11' the measurement. From Ref. [6.5].

Salvetat rt ~1.1' 51 determined expcrimentally the elastic properties from deflection measurements of nanotube ropes with several different diameters. They suspended the ropes as schematically shown in Fig. 6.10 (b), an atomic microscope picture of the experiment is shown in Fig. 6.10(a). Analyzing their results on such ropes as an assembly of beams they deduce an elastic modulus of I TPa, the usual value for nanotubcs. At the same time they determincd the shear niodulus ( M 1 GPa), i.e., the tubes have a high stiffness in the axial direction, but can resist bending only very little. Note that the numerical value of the elastic modulus depends on the diameter of the nanotube assumed. There are differences in thc literature as to how the diameter of isolated tubes and ropes of tubes is given, which does not necessarily reflect different physical constants. Applications based on mechanical reinlorcement through nanotubes should kccp this in mind. Yu ct stretched single-walled nanotubes that they picked up with the tip of an atomic-force microscope from a sample of so-called bucky paper. Compared to multiwall nanotubes they were much more entangled and they could not be stretched across two tips as in Fig. 6.8. In Fig. 6.1 1 wc see the single-walled nanotubc bclore and after the break. In tensile-loading experiments of 15 individual nanotubes, thc authors found again a mean elastic modulus of 1 TPa. Another very interesting application of the elastic properties of carbon nanotubcs was found by Baughman ct a1.1" 531 They developed a micro-mechanical actualor from singlewalled nanotubes, so-called artificialmuscle;\. The idea is to clcctrochemically inject charge on stripes of bucky paper that are close to hut electrically isolated from each other and thus form a capacitor. By applying positive and negative dc voltages to the two sides of the actuator Figure 6.1 1: Scanning electron microscope Image of the tip of an atomic forec microscope pulling on a rope of mglewalled nanotubcl Wetching from a piece of bucky p a p (a) before the break and (h) after the break. From Rcf. r6.521.

h Elastic Properties

Figure 6.12: Micromechanical actuator, a so-called artificial musclc. (a) Different charges on the stripes of bucky paper deflect the strlpc in different direclions. (h) Setup of the electrochemical-actuator cxperiment. (c) Strain or the actuator at different driving frequencies. After Ref. [6.53].

the device as shown in Fig. 6.12 (a) deflected repeatedly up to 1 cm. The actuator was able to follow a square-wave oscillating voltage in the still unoptimized setup of up to 15 Hz, see Fig. 6.12 (c). A smaller device is expected to have higher oscillation frequencies. The maximal strain associated with these deflections was estimated to be -- 0.2%.

6.4 Summary This chapter discussed the elastic and vibrational properties of carbon nanotubes under high hydrostatic pressure. To describe the elaslic response of nanotubes to pressure we developed a continuum model of carbon nanotubes within elasticity theory. The tube was approximated as a hollow graphene cylinder with a finitc wall thickness and closed ends. The linear modulus in the circumferential or radial direction was found to he 2-3 times higher than in the axial direction, i . ~ .the , deformation of the nanotubcs unit cell under pressure is not fully hydrostatic. Instead, a shear strain is present in the graphene hexagons. Because of this nonhydrostatic strain component the E2, graphene optical mode is expectcd to split; its frequency under pressure depends on the dircction of the atomic displacement. For the nanotubes this implies different pressure slopes for axial and circumfcrcntial vibrations, whereas Raman experiments show uniform prcssure dependencies of all high-energy modes. The apparent contradiction is resolved by the phonon eigenvectors with mixed character in chiral nanotubes. Support for this idea comes from ah-initio calculations of the axial and circumferential phonon eigenvectors of two chiral nanotuhes. The calculated atomic displacement of the non-degenerate modes in chiral tubes indeed pointed into various directions. Moreover, the degenerate E eigenvectors even showed a "wobbling" of the displacement direction when going around the circumference. This wobbling is seen as open ellipses in a plot of the z versus the circumferential displacement component, which is a way to uniquely specify a phonon eigenvector in nanotubes. Finally we showed some micro-mechanical applications of nanotubes based on their elastic properties.

7 Raman Scattering

When a new material is discovered - or a long-known material suddenly turns out to be of great physical interest - Rarnan spectroscopy is usually among the first experimental techniques used for characterization. Spectra can be recorded on small and little-known samples, and provide deep insight into the physical properties as well as thc material quality. During the last decade Raman spectrometers have become rather cheap and easy to handle, in particular, the single-grating spectrometers working with a notch filler. The Raman process yields information not only on thc vibrational properties. Resonant scattering is deeply influcnccd by the electronic states of a system, phase transitions are nicely studied by recording the Raman spcctra, and experiments under external pressure allow us to understand the elastic properties as well. In this chapter we briefly discuss the basics of Raman ccattering (Sect. 7.1) in as much as they arc important for the analysis of carbon nanotube spectra, For newly discovered materials without single-crystal quality - such as carbon nanotubes for the time being - one of the interesting questions is what the symmetry of observed modes is. We show in Sect. 7.2 how such a symmetry analysis can be performed on unoricnted samples. In Sect. 7.3 wc discuss how information at large wave vectors can be obtained with Raman scattering. We will pay attention to resonancc phenomena, focusing on the Raman double resonance (Sect. 7.4), which has become a central subject in the understanding of Raman scattering in graphite and carbon nanotubes.

7.1 Raman Basics and Selection Rules First-order Raman scattering is a thrcc-step process as shown in Fig. 7.1. The absorption of an incoming photon with a frequency @ I creates an electron-hole pair, which then scatters inelastically under the emission of a phonon with frequency wph, and finally recombines and emits the scattered photon @.L7 1' Energy and momentum are conserved in thc Raman process 117[7

where the f signs refer to Stokes and anti-Stokes scattering. The details of the electronphoton and the electron-phonon coupling are included in the corresponding matrix elements. Let the polarization of the incoming (outgoing) light be cr ( p ) and the Hamiltonian for electron: electron-phonon coupling is given by H,,. The matrix radiation interaction IJER,= ( H e H p ) the

Figure 4.1: Feynman diagram of a first-order Raman process. An incoming photon with frequency w1 and wavc vector kl excites an electron-hole pair, Thc clectron is scattered inelastically emilling a phonon with frequency w and wavc vector q. The electron-hole pair recombines under the emission of the scattered pholon ( k z ,@). After Ref. [7.1].

element Kzf,,o of the process in Fig. 7.1 is then17,31

where I , O , i ) denotes the state with an incoming photon of energy E l = h w l , the ground state 0 of the phonon (no phonon excited), and the ground electronic state i: the other statcs are labeled accordingly. The initial and final electronic states are assumed to be the same; the sum is over all possible intennediate electronic states a and h. The final phononic state is denoted by f . The EL, are the energy differences between the electronic states n and i; the lifetime of the excited states y is taken to be the same. in Eq. (7.3) and difficult to The Raman intensity is proportional to the square or K2 evaluate, because the matrix clcments are usually not known. For a calculation of the resonant spectrum it is common to take the numerator to be constant and integrate over possible intermediate states u and b as dcscribed, e.g., in Ref. [7.3]. For the double-resonance process another matrix element and a further resonant term in the denominator is included, see Sect. 7.4. In a single (or double) resonance the Raman cross section

I

1

diverges when the intermediate states a or b are real and one (or several) terms in the denominator of Eq. (7.3) vanish. The lifetime y keeps the Raman cross section finite. When interested only in selection rules the question is whether the matrix element in Eq. (7.3) is zero or not. The quickest way to work out the selection rules for Raman scattering in carbon nanotubes is to use the conservation of the quasi-angular momentum m and the parity ~ h (the lattcr only for achiral tubes). Treating m as a conyerved quantum number implicitly see also Chap. 2. This assumes that no Umklapp processes occur in the Raman transitio11;1~.~1 assumption is correct as long as only first-order scattering with q = O and optical transitions in the visible (Ak w 0) are taken into account. An extension to more general transitions, in particular in defect-induced ~cattering,[~.'~ will be performed as well. The eigenstates in Eq. (7.3) are composed of an electronic, a vibronic, and a photonic part. Thc quantum numbers are conserved for the total eigenstate. Sincc the initial and final elec- Am = 0 and o h = +l tronic statc is the same - excluding vibro-electronic coupling[7 for the elcctronic part. A 2-polarized optical transition conserves angular momentum Am = 0 and changes the mirror parity o h = -1; for transitions polarized perpendicular to the tube axis Am = *1 and = $1 .17.81The change in angular momentum and parity induced by the "7[7.71

7.7

117

Ramun Rasics and S d ~ c t i o nRulas

Table 7.1: Phonon symnielries conserving the angular momentum quantum number and the parity in the Kaman configurations. In chiral tubcs oh I S not a symmetry operation; the superscript f and the subscript g are omittcd.

Scattering geometry (z,z)

(44,( w ) , (Y,z), (23.Y) (x,x), (.Y,.Y) (.x,.Y),( ~ 4 )

Phonon symmctries Line group notation Molecular notation of'; A 1, oE; El, oA$, o ~ , t Al,, E2g o@, 0 ~ 2 + A 2,, E2,

absorption and emission of a photon must be compensated for by the phonon. For z-polarized incoming and outgoing light - (zz) configuration in the usual Raman notation - angular momentum and parity are fully conserved by the photons. Thcrcforc, only A: =Al, phonons are = &I allowcd in this scattering configuration. In (xz) or (m) scattering geometry dmphOtOn and o~,,p,,,t,,, = -1 giving rise to E l = E l g phonon scattering. The selection rules for r-point phonons are summarized in Table 7.1 for all possible scattering configurations. Let us now consider the Rarnan selection rulcs for scattering with q # 0, including either defect scattering or a second phonon. Again thcy can be derived in the standard way from the character table (Table 2.5) of the line group[74]~[70] or found from the conscrvation of quantum numbers.' We will see that phonon branches of carbon nanotuhes that arc derived from a Raman-active r-point mode are always allowed. We assume that a possible defect cannot change the symmctry of a state it interacts with (m,lcfc,t = O), ix.,it scatters electrons only within the same band. As for scattering with r-point modes, the phonon provides the quantum numbers by which the electronic system is changed through the optical transitions in a givcn scattering geometry, p.g., Am = mp,, fl for ( x , z ) polarization of incoming and outgoing light, respectively. We summarize thc required quantum numbers in Table 7.2. The 4 parity in achiral tubes (given in bmckets) is defined only for q = 0. The vertical mirror plane (0,) is present for q # O if rn = O or m = n. From the conservation of the 0,parity it follows that the cntirc LO branch (including q = 0) is allowcd in z i g - ~ a gtubes and forbidden in annchair tubcs; and vice V C N U , the TO branch is allowed in armchair tubes but forbidden in zig-zag nanotubcs. In two-phonon scattering, the change in quantum numbers is provided in total by both phonons, i.e., A m = mph = m,~~7,1 +mph,2. Therefore, in contrast to first-order scattering, also modes with Iml > 2 can contribute to the two-phonon signal. They scatter electrons from one band into another band. The quasi-angular momentum m of the phonon defincs to which band the excited electron is scattered through m h = m, m,,h, where the subscripts a and h refer to the intermediate electronic states, see also Eq. (7.24). If the phonon wave vector is larger than ~ / a the , quasi-angular momentum rn is not a conservcd quantum number and the Umklapp rules [Eq. (2.20) and (2.21)] apply. For achiral tubes, m = O changes into m = n for q E (7c/a,2 z / m ] . In chiral tubes, it is more convenient to ; thc entire rii = O band is allowed in ( z ,z ) configuse the helical quantum numbers k,r ~ then uration. To summarize, the selection rules for large-q Raman scattering in carbon nanotubes

-

+

' ~ o t ethat alw the quasi-momentum conscrvation. q , = qz kg,,,,i, round k~rmallytrcm thc full space group character tahle as glven In Tablc 2.5.

Tahle 7.2: Phonon quantum numbers conserving the total quantum numbers in q # 0 Raman scattering. The m quantum number mp), referx to scattering by one or inore phonons. For two-phonon scattering, m,l, = m,h,l -t- m,h,?. If one phonon is replaced by a dcfcct, m,,h = m,h,l f mdccccl = r n p h , l . The parity quantum nurnbcrs given in brackets on the right are only defined at q =. 0. In chiral tubes oh and oxare not a symmetry operation.

Scattering geometry

Allowed phonon quantum numbers

are given by the conservation of the quasi-angular momentum m. If q > K / U , the Umklapp rules must be additionally applied. Form = 0 and m = n phonons in achiral tubes, the allowcd modes are further restricted by parity quantum numbers. In general, for q # 0 the selection rules arc lcss restrictive than for r-point phonons, since somc of the parity quantum numbers are detincd only at q = 0. If a phonon mode is forbidden at the r point but allowed for q > 0, however, continuity prevents the mode from having a largc Raman signal for small q. For an in-depth treatment of thc optical transitions, see Chap. 4. Thc Raman-scattering intensity I is usually evaluated by contracting the Raman tensors '3

where ei and e, are the polarization vectors of incident and scattered light, respectively. The Raman tensors of the modes given in Table 7.1 can be found by the group projcctor technique introduced in Chap. 2. Ofien, an inspection of the table together with symmetry arguments is sufficient to find the general form of the Raman tcnsors. The A 1, representation appears only for parallel polarization?, hence all non-diagonal elemcnts of its Raman tensor must be zero. Various symmetry operations tramform the xx- and the yy-components into each other in nanotubes. To obtain an A 1, rcprc\entation these two entries of the tensor must be the same. The 77-component, on the other hand, is linearly independent, because all line group symmetry operations preserve the z-axis. Wc thus obtain diag[a,a,b], i.e., a purely diagonal matrix with & = a,, = a and a, = b, see Table 7.3, as the general form of an A , , Raman tensor of single-walled nanotubeq. E l g can only have xz, u,yz, zy non-7ero elements. Again xz and yz transform into each other under the symmetry operations of the tubes and are not independent, whereas xz and zx arc dccoupled. Similarly, the other Raman tensors can be found; they are listed in Table 7.3. Thc experimentally obqerved selection rules arise from the 7eros in the Raman ten\or and, with single crystals of carbon nanotubes at hand, the symmetries of all Raman excitations could be determined. At present, however, such crystals are not available. Samples with a partial alignment along the z-axis or dilute nanotube samples on different substrates have been used for Raman e ~ ~ e r i m e n l s . [ ~ 181

7.2 Tensor Invariants When single crystals of a material are not available the symmetries of Raman excitations can he studied by measuring the polarization of thc scattered light in unoriented samples. One can thus determine the so-called Raman tensor invariant^,[^.'^]^[^.^^] which yield information about the symmetry, although the assignment is not always unique. Let us assume a phonon with a diagonal Rarnan tensor with three elements a , # a 2 # u 3 . Furthermore, the scattering configuration in the laboratory frame is (ZZ). To find the Raman intensity we integratc and average over all possible orientations of the crystal. Using Euler's angles17 2'19 [7 221

integrating and rearranging yields

The result in Eq. (7.8) holds for all parallel polarizations of the incoming and outgoing linearly then has a more general form (see polarized light and for every Raman tensor except that below). For perpendicular linear polarization, e.g., Ixz, the integration yields Ixz = 3 y,'2/45. Now it becomes obvious that the symmetry can also he partially deduced from experiments on unoriented materials. For example, if the three elements of the Raman tensor are the same (= a, as for cubic point groups), Raman scaltering is forbidden in crossed linear polarization, and in parallel linear polarization we find simply ILz = u2 and, thus IxZ/IZz = 0. On the other hand, a fully uniaxial Raman lensor (nI = a2 = 0 , a7 = h) results in I x z / l n = 113. In Appendix B we show how to generalize the results of the preceding paragraph for arbitrary Raman tensors. The intensity on unoriented substances follows directly from the

xt2

Table 7.3: Raman tensors of the phonons in carhon nanoluhes. They are valid for the D, and Dgh point groups with q > 3, which refers to all realistic tubes.

7 Raman Scattering

120

transformation of a tensor under rotation. A second-rank tensor can be decomposed with respect to the rotation group into a scalar (tensor of rank zero), an antisymmetric matrix (rank one), and a symmetric traceless matrix (rank two). These irreducible components have wclldefined quantum numbers and transformation properties under rotation. The matrix element for a fixed orientation is obtained from the Wigner-Eckitrt theorem; the integration over all crystal orientations is determined by the tensor invariants. Different authors use slightly different invariants in Raman scattering. Following Ncslor and spiror72" we define the isotropic invariant

the antisymmetric anisotropy

and the symmetric anisotropy

) the elements of the Raman matrix as given in Table 7.3 for carbon where a,, (i, j = x , y , ~ are nanotuhes. The scattering intcnsity on an unoriented sample in any scattering configuration can he expressed by a linear combination of the tensor invariants, see Appendix B. For lincar parallel (11) and perpendicular (I) polarization of the incoming and scattered light the intcnsitics are given by (apart from a constant factor; Table 8.2) Ill = 45a2 +47:

(7.14)

11 = 3x2+5x,2,, (7.1 5) which is the generalized result of Eq. (7.8). The quotient 1 1 / 1 is known as the dcpoliiri~ation ratio p. Under non-resonant conditions the antisymmetric invariant vanishes, y: = 0. Measuring the intensity under parallcl and crossed polarization is then sufficient to find the tensor invariants. In resonance, the antisymmetric scattering does not necessarily vimish, and we need at least one more independent measurement to find the tensor invarianls. Using circular instead of linear polarization the intcnsilies in backscattering configuration are 4

1

I,?c,

~=

= 45a2

+ y,2 + 5 &

(7.16) (7.17)

where IrS0 is the intensity for corotating and I , y ) for contrarotating incoming and outgoing light; I ~ J ~ 3 / l t 1is0 the reversal coefficient P. Solving the system of four equations Eqs. (7.14)(7.17) with respect to the tensor invariants we obtain as one possible solution[721J

45a2

57: 67;

=

2 111-31~y1, I

(7.18)

=

IL-$OO,

(7.19)

=

I(>(?.

(7.20)

With the la5t quantity I(.,(., wc can measure the experimental error. The experimentally obtained tensor invariants are only accepted as significantly different from zero if they are larger = +Id - (Lo,>+I,Y,)l. than In general, the conclusions that can be drawn from experiments on unoriented samples are as follows: The only symmetry that has a non-vanishing isotropic part a is the fully symmetric representation in any point or space group. Thus, phonons of A 1 symmetry are readily distinguished from the other species. Of particular interest is the observation of antisymmetric contributions to the Raman intensity, i.e., # 0. If only the antisymmetric component is present the Raman peak originates from a phonon transforming as the totally antisymmetric representation, in the point group of nanotubcs this is A2 (Table 7 . 3 ) . Totally antisymmetric scattering is scarccly observed experimentally; more li*cquentis an antisymmetric contribution to a degenerate mode.r7,231-17.251 The only possiblc modes in nanotubes showing mixed symmetric and antisymmetric scattering are E l modes. For example, a strong incoming resonance with an optical transition that is allowed in ?-polarization, but forbidden in x- or y-polarization yields in the matrix representation of the El Raman tensor c # d (Table 7.3). Such anti2617[7.271 for multiwall nanotubes. symmetric contributions were reportcd by Rao p t al.17.'2]%L7 Degencrate modes have a symmetric anisotropy y,' different from zero, but a2 = 0. A large ratio $-/a2can serve as an indicator for scattering by El and Ez symmetry modes in carbon nanotubes.

x;,

7.2.1 Polarized Measurements The tensor invariants of the Raman scattered light can be obtained from a linear combination of the intensities in linear and circular polarization as shown in Sect. 7.2. Such a basic setup - a combination of two linear polarizing elements (a Fresnel rhomb and a polarization filter) and a ;1/4 wave plate - is shown in Fig. 7 . 2 . It allows the Raman intensities to bc rcmrded in parallel, perpendicular, corotating, and contrarotating polarization without changing the illumination level or removing any polarizing elements in the light path between the measure-

Fresnel rhomb (90")

NanotubesI

Spectrometer with CCD

A14 wave plate (45")

Analyser (vertical)

Figure 7.2: Raman sctup for the measurements of the tensor invariants. The polarization direction of the incoming light is choscn by the Fresnel rhomb. The laser then passes a ?~/4 zero-order wave plate and i s focused onto the sample. The scattcrcd light comes back through the h / 4 plate, is analyzed with a polarization filter, and focused onto thc cntrance slits of the spectrometer. Different orientations of the Fresnel rhomb and the 114plate yield the relative polarizations without inserting additional elements.

7 Raman Scattering

122

Table 7.4: For a fixed spetrometer polarization the settings or Fresnel rhomb and l.14 wave plate set up as in Pig. 7.2 allow a recording of the four different relative polarizations of incident and scattered light without inserting or removing an optical element.

~ Polarization Parallel Perpendicular Corolating Contrarotating

-

Ill I1

bo Ic3tj

Fresnel rhomb 0" 90" 0" 90"

0" 0"

45" 45"

vertical vertical vertical vertical

ments.[7.19],[7.201,[7 271 We illustrate the arrangement in some more detail since it is relatively unknown; yet it yields the tensor invariants with high accuracy. For the polarizing elements in positions as indicatcd in Fig. 7.2 a backscattering Raman spectrum under corotating polarization is recorded: The laser light is vertically polarized and, after passing the Fresnel rhomb (90" poqition), light is horizontally linearly polarized. After passing the A/4 wave plate (its principle axis is at 45" to the horizontal) the incoming light is circularly polarized. The backscattered beam, in general, consists of a Icft- and a right-hand circularly polarized part, which on passing back through the h / 4 wave plate is converted into linear polarization. The corotating part is now vertically polarized, the contraroting part horizontally (the circular polarizations are specified in the lab frame). Only vertically polarized light can pass the analyzer and is recorded. The intensities in the four different polarizations are then obtained by rotating the Fresnel rhomb and the A/4 wave plate. Let us assume that the analyzer is vertical (e.g.,because of a higher sensitivity of the spectrometer for vertically polarized light), then the Raman polarizations are given by the settings in Table 7.4. Figure 7.3 shows the Raman spectra of CC14 in (a) the two linear and (b) the circular polarizations. In this molecule the fully symmetric mode is characterized by only the isotropic invariant a2being different from zero. For the A1, peak at 460cm-' a depolarization ratio p = 0 and a reversal coefficient P = 0 are expected; the other modes should show p = 0.75 and P = 6 (a2= 0, & = 0). All quantities are in excellent agreement with the measured values given in Fig. 7.3. Deviations from the theoretical value are usually found for the reversal coefficient, because the circular polarizations are much more affected by the non-ideal Figure 7.3: Raman spectra of CC14 (a) under lincar and (b) circular polarization. Next to the fully symmetric modc a1 4h0cm-' and the modc at 3 1 4cm-' the measured depolarization ratio p and reversal coefficient P are given.

Raman shift (cm-')

7.3 Humun Meusurements at L n r ~ ePhonon q

123

backscattering.17"~L72n1 While the theoretical depolarization ratio is the same regardless of thc scattering geometry, the reversal coefficient, e.g., in forward scattering, is the inverse of the bitckscattering value, In CC14 the measured P for the non-fully symmetric modcs varies between 3.6 at 7 ~ 0 c m - Iand 5.3 at 220 cm A more robust indicator for the symmetry of a Raman mode than the raw valucs of p and P are the ratios between the tensor invariants. In particular, the ratio between the symmetric anisotropy and the isotropic invariant "/,2/a2 = 0.4 for Ihc 460 cm-I mode, but y,'/a2 is well above 100 for all other modes.

'.

7.3 Raman Measurements at Large Phonon q The Raman experiments commonly performed in laboratories and excited with laser light in the visible or near-visible energy regime are limited to r-point phonons in first order due to the small wave vector of the incident light. In backscattering geometry the maximal momentum transfer and hence phonon momentum is given by ,q

4n7c

=-

A;

0.01 A-

I

, say at A; = 488 nm

in a material with a refractive index n = 4. This is very small compared to the cxtent of the Brillouin zone in reciprocal space, for which the lattice constants set the relevant scale, k,,

n

= - = 1.3Aa0

I

, say with

ao = 2.46A as for graphite.

(7.22)

There are several techniques of getting to larger wavc vectors that have been used in the past. One is to create an artificial, smaller Brillouin zone by growing superlattices of different ma1(' In a way, this is happening in carbon terials and studying the resulting q = 0 modes.17281317 nanotubes by wrapping up the graphene sheet and thus introducing Brillouin-zone folding, see also Scct. 2.1. Most of these new q = 0 modes in nanotubes are, however, forbidden by selection rules in first-order scattering. A second approach is to relax the perfect periodicity in single crystals. For example, the earliest reports of Raman scattering in graphite by Tuinstra and KoenigL7301 showed that the now-called D mode arose from a defect-induced process at large q vectors; it is not allowed in the perfect crystal. Another example is the ion bombardemcnt of crystalline silicon with Si atoms successively increasing the damage to the material. In this way, one could show how the phonon density of states of a-Si evolved from that of c-Si.17311 The partially oxygenated high-temperature superconductor YBa2Cun07-g with 6 around 0.3 shows defect-induced peaks at 230 and 600 cm-I. Interestingly, the intensity of when illuminated;L7'15L7 "1 light absorption incrrases the order thcsc defect peaks decr~as~s of oxygen atoms in one of the crystal planes (the chain plane). The optical Raman methods of large-q scattering, unfortunately, are connected with little specific information about the phonon dispersions. In second-order Raman ~ c a t t e r i n g , ~341 ~.'~'~~ for example, two phonons arc emitted (or absorbed) at the same time, a process that is generally 10 to 100 times weaker than the first-order spectra. Thc momenta of thc two phonons involved cancel for equal and opposite phonon vectors, and the condition ( q,n, I=I q , - q2 1% 4nn/al is easily fulfilled for visible light throughout the Brillouin zone. As a consequence, the Raman peak is an integration over the Brillouin zone, and regions with large densities of

124

7 Raman Scattering

Figure 7.4: Overbending in carbon nanotubcs as function of chiral angle. The maximal overhending of a given tube in the range 2.8 d 50.OW was fitted to the empirical expression in Eq. (7.23), which is plottcd hcrc.

a<
0, 118 semiconducting tubes, 72 conductance, 85, 87 semiconducting tubes, condition for, 41 Si, 124 single resonance, 127, 145, 163, 172-173

sodium dodecyl sulfate, 72 Stokes scattering, 1 15 strain, 101-1 11 stress, 101-1 11 STS, 47,52,59,61 surface conductivity, 7 1 sword-in-sheath, 112 symmetric anisotropy, 120, 123 symmetry, sre line group, see selection rules, 12-30 phonon eigenvectors, 27-30 Raman modes, 158-159 synchrotron radiation, 125 tensor invariants, 120, 121, 123, 158, 159 tight binding, 3 3 4 1 , 7 1,74 linear hands, 45 ncarest neighbors, 35-38,42 parameters, 36,40 third neighbors, 3 9 4 0 , 4 2 , 5 3 transmission electron microscopy, 9,72 transport electron scattering, 88-93 electronic, 85-1 00 transport measurements, 86 trigonal warping, 44 tuhe-tube interaction, 79 tunneling, 87,93,97 end, bulk etc., 9X twiston, 9 1, 137 Umklupp process, 17, 1 16 unit cell, 4-6 unoriented material, experiments on, 1 19, 158 V shape, 74,75 van-der-Waals interaction, 102, 147 van-Hove singularities, 44,46,59, 73, 80 in Raman, 49 in STS, 47,59

wave function, 57 wave vector allowed of a tube, 8,43 Wigner-Eckart theorem, 120 work function. 88 X-ray monochromator, 125 xy-polarized light, 69

Index

Young's modulus, 102 z-polarized lighl, 69 zeolite crystals

filling fraction, 78 nanotubes in, 7 1,77, 144 zone folding, X,4144,53, 77, 123, 136, 160 phonon dispersion, 125, 117